spherical curvature

7/28/2019 Spherical Curvature

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Russian Math. Surveys 50:1 1-68 1995 BL Board and LMSUspekhi Mat. Nauk 50:1 3-68 UDC 514

The geometry of spherical curvesand the algebra of quaternions

V. I. Arnol'd

ContentsIntroduction 2Chapter I. Thegeometry of curves onS2 3

1. Theelementary geometry of smooth curves and wavefronts 32. Contact manifolds, their Legendrian submanifolds and their

fronts 9 3. Dual curves and derivative curves of fronts 104. Thecaustic and the derivatives of fronts 12

Chapter II. Quaternions and the triality theorem 135. Quaternions and the standard contact structures on the

sphere S3 136. Quaternions and contact elements of the sphere 5? 157. The action of quaternions on the contact elements of the

sphere 5| 188. Theaction of right shifts on left-invariant fields 209. Theduality of j-fronts and fc-frontsof -Legendrian curves 20

Chapter III.Quaternions and curvatures 22 10. Thespherical radii of curvature of fronts 22 11. Quaternions and caustics 23 12. Thegeodesic curvature of the derivative curve 24 13. Thederivative of a small curve and thederivative of curvature

of thecurve 28Chapter IV. Thecharacteristic chain and spherical indices of a hyper-

surface 30 14. Thecharacteristic 2-chain 31 15. Theindices of hypersurfaces on a sphere 33 16. Indices as linking coefficients 35 17. Theindices ofhypersurfaces on a sphere asintersection indices 36 18. Proofs of the index theorems 38 19. The indices of fronts of Legendrian submanifolds on an even-dimensional sphere 40

Chapter V. Exact Lagrangian curves on a sphere and their Maslovindices 44 20. Exact Lagrangian curves and their Legendrian lifts 45


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V. I. Arnol'd 21. The integral of a horizontal form as the area of the character-

istic chain 4822. A horizontal contact form as a Levi-Civita connection and a

generalized Gauss-Bonnet formula 49 23. Proof of the formula for the Maslov index 52 24. The area-length duality 5425. The parities of fronts and caustics 56

Chapter VI. The Bennequin invariant and the spherical invariant J+ 57 26. The spherical invariant J+ 58 27. The topological meaning of the invariant SJ+ 59

Chapter VII. Pseudo-functions 6028. The quasi-functions of Chekanov 61 29. From quasi-functions on the cylinder to pseudo-functions on

the sphere, and conversely 62 30. Conjectures concerning pseudo-functions 6331. Space curves and Sturm's theorem 66

Bibliography 68

IntroductionThe spreading of a wave over the surface of a two-dimensional sphere may be

described by means of various curveswave fronts, rays, caustics, and so on.A system of wave fronts is a system of equidistants of a single curve. Even if this

curve is smooth the wave fronts in general have singularities. The singularities ofall the wave fronts of the system themselves form a curve, called the caustic. Thecaustic in general has cusps, just as the wave fronts do.

Another description of the spreading of a wave is given by a Lagrangian curve.Such a curve divides the area of the sphere in half but in general has no singularitiesapart from self-intersections.

The connections between all these curves, their singularities and their topologicalinvariants are clarified by means of three natural complex structures of quaternionspace: namely the connections between three projections of the same Legendrianknot along three Hopf bundles S3 -t S2.

The following are studied below as consequences of these connections:1) several generalizations of the classical four-vertex theorem for a plane curve;2) an unusual version of the Gauss-Bonnet formula for an immersed circle in the

sphere and for wave fronts;3) an explicit formula for the Maslov index of a Lagrangian curve on the 2-sphere;4) unexpected connections between the indices of points with respect to hyper-

surfaces on even-dimensional spheres;5) conformal invariants of an immersed circle in the plane that generalize the

Bennequin invariant;6) newpartly proved and partly conjecturedgeneralizations of the Morseinequalities and theorems of Chekanov and Givental' on Lagrangianintersections and Legendrian links in symplectic and contact topology;

7) theorems on a duality between area and length in spherical geometry.


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T h e geometry of spherical curves and the algebra of quaternions 3In a natural way all these facts arise directly from consideration of the influence

exerted by the symmetry of the quaternion units on the symplectic geometry of the2-sphere and the contact geometry of the 3- sphere. Without these connections thefacts stated might have remained unnoticed for a long time.Since, however, the quaternions can be banished from the formulation of thetheorems in question (and, if it were desired, from their proof), I begin straightaway with their formulation (the motivation of which, for the most part, lies in thesymmetry of the quaternions).

CHAPTER ITHE GEOMETRY OF CURVES ON S2

We present here geometrical constructions which relate caustics and wave frontson the 2- sphere. The meaning of these constructions would be clearer if consideredfrom the point of view of contact and symplectic geometry. However I begin withvery elementary formulations, which do not require any prior knowledge for theirunderstanding.

1. The elementary geometry of smooth curves and wave frontsWith every curve on the 2-sphere of radius 1 we associate two other curves. The

first of these is obtained by moving each point of the curve a distance / 2 alongnormals to the original curve. This curve consists of the centres of curvature ofgreat circles of the sphere tangent to the original curve. For this reason it is saidto be dual to the original curve (Fig. 1).

Figure 1

Definition. The curve dual to a given co-oriented curve on the sphere is the curveobtained from the original curve by moving a distance / 2 along the normals onth e side determined by the co- orientation.This definition applies not only to smoothly immersed curves, but also to wave

fronts, having cusps (of semicubical type or, in general, of type xa = y a+ 1) .


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4 V. I. Arnol'd

T h e dual curve itself is naturally co-oriented and is a wave front equidistant fromth e original one (lying at a distance / 2 from it).T h e cusps on the original front correspond to points of inflection on the dualfront, while points of inflection on the original one correspond to cusps on the dual(a point of inflection of a spherical curve is a point having an order of contactgreater than one with the tangent great circle).T h e second dual of a front is antipodal to the original one, while the fourthcoincides with the original one.Definition. The derivativeof a co-oriented curve on the oriented standard sphereS2 is the curve obtained by moving each point a distance / 2 along the great circletangent to the original curve at that point (Fig. 2). The direction of motion alongth e tangent is chosen so that the orientation of the sphere, given by the directionof the co-orienting normal and the direction of the tangent, is positive.

Figure 2This definiton applies also not only to smoothly immersed curves, but also to

wave fronts.1Example. On the unit sphere the parallel of latitude of Euclidean radius cos isdual to the parallel of latt itude of Euclidean radius sin# . The length of the firstparallel is equal to the area between the second parallel and the equator of thesphere.

T h e derivative of any parallel of latitude is the equator parallel to it.I t turns out that the coincidence of length and area and the division by theequator of the sphere into halves is not accidental.Theorem 1. The derivative of a wave front coincides with the derivative of any ofits equidistants and is a smoothly immersed curve on S2 even if the original wavefront has generic singularities.

1 Derivatives of smoothly immersed curves are studied in a recent work by B. Solomon(B . Solomon, Tantrices of spherical curves, P rep r i n t , University of Ind iana , 1993) t h a t refers toearlier results of Jacobi and Kagan. The author is grateful to S.L. Tabachnikov for communicatingth i s preprint.


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T h e geometry of spherical curves and the algebra of quaternions 5Remark. In the definition of the derivative the movement along tangents by a dis-tance / 2 may be changed to moving by an arbitrary constant distance s. The curveso obtained is also smoothly immersed in S2, provided that s is not a multiple of (as B.A. Khesin has shown to me).F or a plane wave front one can move all the points along tangents by an arbitrarynon-zero constant distance: one always obtains a non- singular (smoothly immersed)curve.T h e derivative curve of a closed wave front is not an arbitrary curve immersedin the sphere: it satisfies a topological condition of quantization.

If the derivative curve has no points of self- intersection, then this conditionconsists simply in that, just like the equator, it divides the area of the sphere intotwo equal parts.On the other hand if the derivative curve of a closed co-oriented front has pointsof self- intersection, then the condition of quantization is given by the formula

th e area bounded by the derivative curve = ,where is the Maslov index of the original wave front. The Maslov index of awave front, defined in [1], is equal to the algebraic number of cusps of the front.In calculating a semi-cubical cusp is considered to be positive if the chords thatjoin points of the branch approaching the cusp to points of the branch leaving itco-orient the curve positively in a neighbourhood of the cusp. The number sodefined is the Maslov index of the closed curve that corresponds to the front on thetwo- dimensional Lagrangian manifold of the cotangent bundle of the sphere, thatis, on the symplectization of the front (see [1]).

To clarify the quantization condition it remains to define 'the area bounded byan immersed curve on 5 2 ' . Of course, this area is the integral of the standard areaform over a 2-chain, whose boundary is the given immersed curve in S2.

However the 2-chain defined by this condition is not unique, but only moduloth e 2-cycle S2. So 'the area bounded by an immersed curve in S2' is defined onlymodulo 4 .

I t turns out, however, that among all the 2-chains bounded by a given curveimmersed in the 2- sphere there is a distinguished 2-chain (called below the charac-teristic chain).T h e characteristic chain appears in an analogue of the Gauss- Bonnet formula forcurves immersed in S2. There is an analogous construction also for hypersurfaceson all even- dimensional spheres.T h e geodesic curvature of a co-oriented curve is considered to be positive if thecurve turns away from its geodesic tangent on the side of the co- orienting normal.T h e o r e m 2. The integralof the geodesiccurvatureof a closed co-oriented curveimmersed in S2 is equalto the area of its characteristic 2- chain.

T h e integral of the area of this characteristic 2-chain for an arbitrary closedco- oriented front is equal to , where is the Maslov index of the front.T h e co- orientation of the derivative curve is chosen here so that the co-orientingvector together with the orientation form a positively oriented frame of S2. Such aco- orientat ion is said to be correct.


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6 V. I. Arnol'd

I now introduce a formula for calculating the characteristic chain of a co-orientedhypersurface on an oriented even- dimensional sphere.

This formula has the form of the product of the volume element of the sphere bya locally-constant function on the complement of the hypersurface. This locally-cons tant function depends only on the co- orientation of the hypersurface and doesn ot depend either on the orientation of the hypersurface or the orientation of thesphere. On change of sign of the co- orientation of the hypersurface this functionchanges sign.

We select one of the points of the complement of the hypersurface on the sphere.We call this the 'point at infinity' and denote it by B.

Stereographic projection maps the complement to the point to an orientedEuclidean space. The hypersurface projects to a compact co-oriented hypersurfacelying in this space. This hypersurface of the Euclidean space has a certain index is( the degree of the Gaussian map to a sphere). Moreover, for any point A of thecomplement to this hypersurface there is denned the winding number i% of thehypersurface around A (the degree of the map of the hypersurface to a sphere,sending a point C to the direction of the vector AC).Theorem 3. The difference i^ ( / 2) is conformally invariant (that is, itdependsonly on the point A and doesnot dependon B, the chosen point at infinity).

This difference, considered as a multiplicity function of the point A (integrallyor semi- integrally valued), turns the complement to the original hypersurface intoa chain whose boundary is the original hypersurface. This is the chain called thecharacteristicchain of the original hypersurface.T h e conformal invariant of the winding number of the hypersurface A (correctedby subtracting is/ 2) is explained by the following elementary (but apparently notpreviously observed) identity.Theorem 4. There is a formula connectingthe indices of points with respectto ahypersurfaceand the indices of the hypersurface in even- dimensional space, namely

2iBA = iB - -Consider now a closed co- oriented wave front on S2 with Maslov index , its

correctly co- oriented derivative curve with length element dl, and its characteristic2- chain c (so that dc = ).Theorem 5. The integral of the area form over the characteristicchain, theMaslovindex of the originalwave front and the geodesic curvature of itsderivativeare connectedby therelations

= / .

Remark. The equality of the left and right parts of these equalities is a generalformula of Gauss-Bonnet type, satisfied by any closed immersed curve (not nec-essarily occurring as the derivative of a closed front).

A similar formula of Gauss- Bonnet type holds also for hypersurfaces immersedin a sphere of arbitrary even dimension.


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T h e geometry of spherical curves and the algebra of quaternions 7T he reason for the connection between the derivative curve and the Maslov

index is the following, also elementary, fact concerning the geometry of curves onth e standard two- dimensional Riemannian sphere.Let us recall that the derivative curve is obtained from the original wave fronton the sphere of radius 1 by moving its points a distance / 2 along the tangent

great circles.Theorem 6. The directionsof these great circlesat points of the derivative curveare parallel in the sense of the standardRiemannian geometry of the sphere S2.

T h e preceding theorems allow one to determine necessary and sufficient condi-tions on a curve immersed in S2 for it to be the derivative of a closed co- orientedwave front: the area of its characteristic chain must be a multiple of 2 .

In particular, a curve embedded in S2 is the derivative of a closed co- orientedfront if and only if it divides the sphere into two parts of equal area (like an equatorof the sphere).2T h e cusps of a system of fronts equidistant from one another form the caustic(Fig. 3).

Figure 3

Theorem 7. The causticof a system of equidistants from a closed co- oriented wavefront on the standard two- dimensional Riemannian sphere is dual to the commonderivativecurve of any of the fronts equidistantto one another that form the family.

Remark. In particular, the caustic does not have any points of inflection, since itsdual is the derivative curve of fronts smoothly immersed in the sphere and so hasn o cusps.

2 T h e derivative of a smoothly immersed closed curve has full geodesic curvature equal to 0an d does not have an arc of full geodesic curvature ; each closed smoothly immersed curve in thesphere with these propert ies is the derivative of a smoothly immersed closed curve, as B. Solomonhas shown in the work cited earlier.


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8 V. I. Arnol'dTheorem 8. The points of inflection of a family of fronts equidistant from oneanother lie on two smoothly immersed curves in S2, each of which bounds an areawhich is a multiple of 2 , these being the derivative curves of the caustic.

I n fact the points of inflection of a front lie at a distance / 2 from the cusps ofthe dual front along the great circle orthogonal to both fronts.

At a cusp of a front this great circle of the sphere is tangent to the caustic.Therefore the points of inflection of all the fronts of the family lie on the pair ofderivative curves of the caustic (furnished, for each of the two curves of the pair,with one of the two possible co- orientations).

The topological restriction on the area bounded by the derivative curve is trans-formed on transfer to the dual curves into a quantization condition on the lengthof the caustic.Definition. The oriented length of a generic caustic is the alternating sum of thelengths of its segments between successive cusps.Theorem 9. The oriented length of the caustic of a system of closed equidistantwave fronts on the standard Riemannian two- dimensional sphere is equal to anintegral multiple of 2 (in fact equal to , where is the Maslov index of anyfront of the family).

Which of these segments of the caustic to be counted here as positive will besettled in 24.Theorem 9 generalizes a known property of the caustic of a closed plane curve(the oriented length of such a caustic, for example the astroid, arising as the causticof an ellipse, is equal to zero).

The stated property is explained by the fact that the front is obtained from thecaustic 'by unwinding a thread from it' (just as an evolvent from its evolute). Theincrease in the length of the free part of the thread is equal to the length of thethread unwound from the caustic, so long as the caustic is convex (Fig. 4).

Figure 4


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T h e geometry of spherical curves and the algebra of quaternions 9On passing through a cusp of the caustic the direction of convexity changes. It

follows that the length of the unwound thread after such a point should be countedwith opposite sign.

If the evolvent of a plane curve is closed, then, for a complete circuit of it, theincrease in length of the free end of the thread must be equal to zero. Therefore thealgebraic sum of the lengths of the segments of the plane caustic between successivecusps (with alternating signs) must be equal to zero.On the sphere this sum is not zero, but is a multiple of 2 , since the returnto the original point of the evolvent no longer requires the return of the lengthof the free part of the thread to its original value. The increase of this length by2 corresponds to adding to the free part of the thread a complete circuit of thesphere along the great circle containing the free part of the thread. In that caseth e increase of the length of the free part of the thread by an integral multiple of2 does not change the end point of the thread.T h e preceding theorem shows that this increase of the length of the free part ofth e thread after a full circuit of the caustic is equal to , where is the Maslovindex of the front (which for a co- oriented front is always even).Remark. The equality of the area of the characteristic chain of a closed curveimmersed in the sphere and the oriented length of its dual curve is independentof the fact that the immersed curve is a derivative: this is a general 'area- lengthduality' in the geometry of S2 (the sign of the length changes on passage througheach cusp of a generic front).3To formulate all these formulae with their boring signs with a reasonable degreeof generality I must first of all recall some definitions from the general theory ofwavefronts and caustics in contact and symplectic geometry. After this, general theoremswill be proved (with the aid of quaternions). From these the theorems of this sectionwill follow. Theorem 1 is proved in 6 and 9 of Chapter II. Theorems 2, 5 and 9are proved in 22, 23 and 24 of Chapter V Theorems 3 and 4 are proved in 18of Chapter IV Theorems 7 and 8 are proved in 4. Theorem 6 is proved in 12 ofChap t e r III. 2. Contact manifolds, their Legendrian submanifolds and their fronts

A linear hyperplane E"1 of a tangent space to an - dimensional smooth man-ifold Mn is said to be a contact element of Mn. A co- orientation of the contactelement is the choice of one of the two halves into which it divides the tangent space.A co- orientation of a contact element of a Riemannian manifold is determined bya choice of direction on the line normal to it.

All the (co- oriented) contact elements on Mn form the bundle space 2 ~ of asmooth fibration

p: ST*Mn -> Mnwith fibre S"'1 (the cotangent sphere bundle of Mn). This manifold E2n~l isequipped with the natural 'tautological' field of tangent hyperplanes. The hyper-plane of the tautological field at a point s of E2n~1 is the inverse image by p, of

3 T h e area- length duality was encountered earlier by Santalo in works on integral geometry; ithas recently been employed by S.L. Tabachnikov in Russian M a t h . Surveys 48:6 (1993), 81- 109an d by J.C. Alvares (Rutgers University, 1994).


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10 V. I. Arnol'd

th e hyperplane in the space tangent to Mn, which is represented by the point s ofth e manifold E2"- 1.T h e tautological field of hyperplanes on E2n~x is called the naturalcontact struc-ture of the manifold of contact elements on M".Example. The manifold of co-oriented contact elements of S2 is the projectivespace EP 3 = S3/ 1. Its natural contact structure is obtained from the field ofplanes orthogonal to the fibres of the Hopf bundle S3 - > S2, under the projectionS 3 - S3/ 1.

T h e identification S T * S 2 S3/ 1 will be described in detail below, in 6.Definition 1. A Legendriancurve in a three-dimensional contact manifold E3 isan integral curve of the contact structure, that is, an immersed curve in 3, whosetangent at each point lies in the plane of the contact field of planes.4

Example. Each curve immersed in a surface M2 determines a Legendrian curvein the contact manifold E3 = ST*M2. This curve consists of the contact elementson M2, tangent to .A point on M2 also determines a Legendrian curve in E3. This curve consists

of the contact elements of M2, applied at the point (that is, it is a fibre of thecotangent sphere bundle).Definition 2. The front of a Legendrian curve L : Sl -> E3 on the manifold E3of contact elements of a surface M2 is the projection of this Legendrian curve ontoth e surface pLiS1) C M2.

T h e front of a generic Legendrian curve on the surface has semicubical cusps andpoints of transversal self- intersection as its only singularities.T h e contact elements forming the original Legendrian curve determine aco-orientationof that front, namely a concordant choice of normal to the front(at double points of the front there are two normals).

Such co-oriented fronts of generic Legendrian curves on the 2-sphere will beth e starting material of our constructions. 3. Dual curves and derivative curves of fronts

We consider a co- oriented front in an oriented sphere S2 of radius 1.T h e great circles of the sphere, orthogonal to the front, will be called its rays.T h e rays are oriented (by the co-orientation of the front).On moving each point of the front along the ray by a distance t we obtain a newcurve *, called the t- equidistant of the front. For example, 2 = .

According to the Huygens' principle, an equidistant is orthogonal to the raysemanating from the original front. An equidistant co-oriented by the directions ofthese rays is the front of a Legendrian curve, diffeomorophic (and contactomorphic)to the original one.

4 A Legendrian submanifold of the contact manifold E2n l is an integral submanifold of max-imal possible dimension (equal to 1).


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The geometry of spherical curves and the algebra of quatern ions 11

Definition 1. The front equidistant by / 2 from the original one is said to bedual to .5

Example. The curve dual to a parallel of Euclidean radius sin on S2 is a parallelof Euclidean radius cos#.T h e following lemma (clarifying the connection with projective duality) is

obvious.Lemma 1. The dual front is formed from the centres of the great circles tangentto the original curve. The second dual of a front is antipodalto the originalone:pVV _ p

In particular, points of inflection of the original front correspond to cusps of thedual (and conversely).T h e co- orientation of the original front on the oriented sphere determines atevery point of the front (including singularities) a positive tangent direction (sucht h a t the frame determined by the co- orientation and orientation directions orientsth e sphere positively).Definition 2. The derivative curve " of a front is formed from those points ofth e great circles tangent to which lie at a distance / 2 from the point of tangencyon the positive side.Example. The derivative of a parallel of latitude of the sphere is its equator.Remark. The derivative curve may be considered to be the curve of normals in thesame way that the dual curve may be considered to be the curve of tangents.T h e following lemma (clarifying the term 'derivative') is almost obvious.Lemma 2. The derivative of a front coincides with the derivativeof an arbitraryfront equidistantfrom it.Example. ( ) ' = ': the derivatives of dual fronts coincide. The derivatives ofanitpodal fronts also coincide: ( )' = '. Change of the orientation of a frontchanges the sign of the derivative.Theorem 1. The derivativeof the front of a genericLegendrian curve is asmoothcurve (immersed in the sphere), bounding the area , a multiple / 2 .

T h e topological meaning of the even number appearing here (in fact the Maslovindex) will be considered in Chapter VTheorem 2. The great circles tangent to a given front form at the points of thederivative curve of this front a framing, parallel in the sense of the Riemanniangeometry of thesphere.

T h e connection between the integral of the geodesic curvature of the derivativecurve and the Maslov index of the front will be discussed below in Chapter V5Th e differential geometry of a pair of dual hypersurfaces on spheres has been a classical object

of study (see, in particular, Chapter 17 of Santalo's book); it is remarkable that here it is stillpossible to find new results, even for curves on the two- dimensional sphere.


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12 V. I. Arnol'd

Theorem 3. The anglebetween the great circlestangent to two equidistant frontsat the point of their intersection on the derivative curve is equal to the distancebetween the equidistants(and, in particular, is constant alongthe derivativecurve).

Theorem 3 is obvious.T h e proof of Theorems 1 and 2 will be given below in 18 and 12.

4. The caustic and the derivatives of frontsConsider the family of all the equidistants of a given co-oriented front on the

standard sphere. Even if the original front is non- singular, some of its equidistantswill have singularities.

A singular point of a front is a critical value of the projection of the correspondingLegendrian curve.T h e singular points of generic fronts are semicubical cusps. The double pointsof a front in general are not singular.Definition 1. The caustic of a family of fronts equidistant from one another is thecurve formed from their singular points.

T h e caustic of a generic front has as its only singularities semicubical cusps andpoints of transversal self- intersection.Theorem 1. The derivativeof a front is dual to its caustic.Proof.The front at a cusp is orthogonal to the caustic. Therefore to move sucha point a distance / 2 along the tangent to the front is the same as moving it adistance / 2 along the normal to the caustic (Fig. 5).

Figure 5

Remark 1. The caustic, in general, is not co-oriented. In this theorem theco-orientat ion of the front is not taken into account. Therefore the derivative of thefront in Theorem 1 is understood to be the union of both (mutuallyantipodal) components, obtained by moving along the tangent a distance / 2 ineither direction.


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T h e geometry of spherical curves and the algebra of quaternions 13Remark 2. The caustic of a co-oriented front on the sphere consists of two different(mutually antipodal) components. This follows easily from the evenness of thenumber of cusps of a co-oriented curve on the sphere.Corollary. The caustic of a family of fronts on the sphere equidistantfrom oneanother has no points of inflection.

In fact, its dual, being the derivative of any of the fronts equidistant from oneano the r , has no cusps (by Theorem 1 of 3).Theorem 2. The points of inflection of all the fronts equidistant from a given oneform an immersed curve on the sphere, boundingan area that is a multiple of 2(namely the derivativeof the caustic, which is consequently the secondderivative ofthe front).Proof.The points of inflection of the front are obtained from the cusps of the dualfront (lying on the caustic) by moving a distance / 2 along the ray orthogonal toth e front at the points of inflection and the cusps. At a cusp it is tangent to thecaustic, and this proves Theorem 2 (in view of Theorem 1 of 3).

CHAPTER IIQUATERNION S AND THE TRIALITY THEOREM

Passing from the wave front to the dual front and to the derivative curve of themboth corresponds in algebra to passing from the imaginary unit i to the imaginaryunits j and k in the space of quaternions (see 9).In order to explain this we need some elementary facts from the geometry ofquaternions, set out in 5-8. Basic to these facts is the easily proved identityekselte~ks = e*1, = i cos 2s + j sin 2s

(where s and t are real). The other computations of this chapter are just as simple,but it is necessary to carry them out in order not to get lost among numerous signsand orientations. 5. Quaternions and the standard contact structures on the sphere S3

Let be an imaginary quaternion of length 1 (for example, i, j or k). Mul-tiplication of all the quaternions by on the right imparts a complex structure(depending on ) to the space of quaternions R 4.We associate with each quaternion q the operator of multiplication on the leftby q in the space of quaternions R 4.Proposition 1. The operatorof multiplication by the quaternion q on the left is - complex.Proof.In fact, denoting quaternion multiplication by a dot we have q (z ) =(q ) for every quaternion z.


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14 V. I. Arnol'dProposition 2. The determinant of the operatorof multiplicationon the left by aquaternion of length 1 is equal to1.Proof.Both length and determinant aremultiplicative. Therefore the image ofthem ap det : S 3 - C\0 is a compact connected subgroup. Theconnected compactsubgroups of C\0 are two in number: the point 1 and S 1. But S 3 does not fibreover S 1. Accordingly, det(53) = 1.Corollary. Thedeterminant of the - complex operator of multiplication on theleft by an arbitrary quaternionis equal to thesquareof itslength.Proof. det(g) = det(|9|) det(g/ |g|) = |9 |2 1,since the determinant of the operator of multiplication of a vector in the complexplane by a real number is equal to thesquare of that number.Example. The vectors 1 and j form an i-complex basis for E 4 . This means thatth e quaternion x+yi+uj+vk may be considered as the complex vector (x+iy, uiv)(pay attention to the sign!).

With respect to this basis the operator ofmultiplication on the left by the quater-nion q = a + bi + cj + dk has thematrixfa + bi c di\ (A C\ , . , . 2 , i2 , 2 , .2\ c - di a bij \C A J

Accordingly, thegroup S 3 of quaternions of length 1 is isomorphic to thegroupSU(2) of unitary matrices of degree 2 with determinant 1.If we were to start from another complex structure involving instead of i weshould obtain another isomorphism S 3 -> SU(2).F or each quaternion of length 1 consider now its corresponding Hopf bundle

associating with each non- null point of the >r-complex line passing through 0 thedirection of that line.T h e fibre of this principal S 1 - bundle is thecircle

{z-e** : t M/ (2TTZ )} ,where 6 S 3 is a point of the fibre. Theaction by S 1 on the fibre is determinedby shifts of the real axis t.Warning. Our notations are not complex, but quaternionic. If is the operatorof the complex structure on C2 such that

I*z = ,then z = eI"tz.

Consider on the sphere S 3 ofquaternionsofunit length the vector field , tangentto the fibres of the bundle ,


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The geometry of spherical curves and the algebra of quaternions 15Proposition 3. The vector field is left- invariant (under the action of the groupS3 of quaternions of length 1 on itself, by multiplication by elements of S 3 on theleft).Proof.

g ( z e*) = ( ? * ) e " 4 .

Consider th e field of planes on S 3 orthogonal to the vectors of the field . Thisfield provides 5 3 with a contact structure, which we call th e >r- structure.Proposition 4. Thecontact - structure is left- invariant (with respect to the actionof the group S 3 of quaternions of length 1 on itself by multiplying elements of S3 onthe left) and gives an S3- left- invariant connection to the principal S 1 - bundle .Proof. Left shifts preserve th e metric on S 3 and the vector field , and thereforealso th e field of planes orthogonal to it . The S 1 - structure of the fibres is alsopreserved, since it is given by th e parameter t.

We have constructed on S 3, in particular, three 53- left- invariant vector fieldsi,j,k and the contact structures that correspond to them and determine S3- left-invariant connections on the three principal Hopf bundles ; : S 3 - > S 2, , : 5 3 ->5? , an d 7rk : S 3 - > S 2.

6. Quaternions and contact elements of the sphere S2Suppose that t h e i- contact structu re is assigned to the space of the j- bundle TTJ

constructed in 5.Theorem 1. The bundle T CJ : S 3 - S | is naturally isomorphic to the doublecovering over the bundle of co-oriented contact elements of the 2- sphere,TTJ : ( S T * S 2 E P 3 ) > S 2. Under this isomorphism the contact structure onS3 projects to the natural contact structure on the space of contact elements.Proof of Theorem 1. We begin with the following obvious remark.Lemma 1. The fibres of the bundle TTJ are Legendrian (that is, simply integral)curves of the i- structure.I n fact, th e fields i and j are orthogonal at every point. Therefore the direction oft h e fibre of j is tangent to the orthogonal i- plane of the contact i- structure.

Thus th e two- dimensional planes of th e contact i- structure project along th efibres of the j- bun dle to one- dimensional contact elements of the sphere Sj. Theseelements are co- oriented (by th e projection of th e direction of the field i).Definition. Th e co- oriented contac t element of th e sphere S1? corresponding to agiven plane of the contact i- structure on S 3 is theprojection of the plane co- orientedby th e vector i along th e fibres of the j- bundle.

We prove that th e constructed map S 3 >S T * S ? is a double covering and hast h e properties stated in th e theorem.


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16 V I. Arnol'd

Lemma 2. On moving a point on S2 with velocity 1 alonga fibre of the j- bundlethe contact i- plane changesin such a way that its projectionon the sphereS2 turnswith angular velocity2.

Proof. We compare the directions of the vectors of the field i at points and e7'*of a fibre of the j- bundle. We bring the vector e^ i of this field from the point ejt back to the point by acting on the right with the quaternion e~jt (whichdoes not change the projection onto S2). We obtain at the point the vectorz- ei'- i- e- i*.

On varying t this vector turns in the plane of the vectors at of the fields i and k,orthogonal to the direction j of the fibre. The rotation is from the direction i toth e direction k, and has (constant) angular velocity 2. For example, for small t

ejt i e~jt ss i + (ji - ij)t +

From Lemma 2 it follows that after the time t = 2 of a complete turn alongth e fibre of the j- bundle on S 3 the corresponding contact element on S'j has turnedthrough an angle 4 . Consequently, our natural map S3 - > S T * S ? is fiber wise adouble covering of principal ^- bundles.

T h e projection of the contact i- structure by this map is the field of planes gen-erated by the projections of the vector fields j and k. Such a plane contains thevelocities of motion of the contact elements for which the velocity of the point ofapplication belongs to the element (since the projection of the vector k onto thesphere belongs to the element, while the direction j , by Lemma 2, corresponds toth e rotation of the element).T he constructed double covering S 3 >S T * S j allows one to identify the latterspace in a natural way with the factor-space EP 3 = S3/ 1. In fact, e


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T h e geometry of spherical curves and the algebra of quaternions 17

Figure 6

The triality theorem. The projection TT^L is congruent to the front dual to ,while the projection T T I L is congruent to the derivative " / .

I n this way a front, the dual front, and the derivative of them both, are threepersonifications of one and the same Legendrian curve (F ig. 6).

T h e next few sections are dedicated to th e proof of th e triality theorem, com-pleted below in 9. The proof establishes also explicit forms of the isometriesS% Sj an d Sf > Sj th at realize the congruences stated in the theorem.

T o be precise, the first is induced by multiplication of S3 on th e right by e I 7r / 4,a n d the second by multiplication by e fc7r/ 4.

F r o m the triality theorem it follows that the derivative curve is smoothlyimmersed. In fact the i- Legendrian curve L, smoothly immersed by definition,is transversal to the direction of the fibres of th e i- bundle. Therefore its projection ! on the sphere Sf also is smoothly immersed. So Theorems 1 of 1 and 3follow from the triality theorem.

Remark. The proof of the smoothness of the curves obtained from a front by movingalong tan gent s by a distance 2s th at is not a multiple of is similar to the proof oft h e smoothness of the dual curve given below, except that multiplication by e*71"/ 4has to be replaced by multiplication by eks.


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18 V. I. Arnol'd

7. The action of quaternions on thecontact elements of the sphere S?

Let 53/ 1 be identified with the space S T * S ? of co- oriented contact elementsof the sphere 5 | by means of the complex structure j and the contact structure i,

as has been described in 6. Let t be a real parameter.T h e action of multiplication on the right by the quaternions elt,ejt and ekt onS3 induces on S 3 / l S T * S ? three dynamical systems, described in the followinglemmas (Fig. 7).

Figure 7

Lemma 1. Under the action of elt on ST'S ' j a contact element moves alongageodesic at all times orthogonal to it in the direction of its co- orientation withspeed 2.Lemma 2. Under the action of eJ< on S T * S ? a contact element turnsaroundits fixed point of application with angular velocity 2 in the directiongiven by thej-complex structure on the sphereS?.Lemma 3. Underthe action of ekt on S T* S " j a contactelementmoveswith speed 2along a geodesic at all times tangential to it in the direction obtainedfrom theco- orientation directionby turning throughan angle / 2 in the directiongiven bythe j- complex structure on the sphere5?.Proof.The base 5? of the bundle , is obtained from S3 by factorizing it by rightshifts (by elements of the form eJ>t). Therefore left shifts by elements of S3 act onth e factor- space Sj.

These left shifts preserve the eJi- right- invariant metric on S3 and, consequently,preserve the natural metric on the sphere S?. Since the sphere 5

3is connected, theleft shifts act on the sphere S? as motions.

All motions of the sphere 5 | are obtainable in this way from left shifts on S3,since any element of S3 can be carried by left shifts into any other, and that meanst h a t any direction tangent to the sphere 5? may be carried into any other.


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T h e geometry of spherical curves and the algebra of quaternions 19Consequently, it is enough to verify Lemmas 1-3 for a specially chosen elementof S3 and it is natural to choose the quaternion 1.

Lemma 4. Let be an imaginary quaternion of length 1, orthogonal to j . Thenthe projection , maps the great circle{e*1 : t mod 2 } of the sphereS3 ontoa great circleof the sphereSj as a double covering.Proof of Lemma 4. Consider the circle of quaternions cost + xsint, e l ,as a curve in the j- complex plane. Choose in this plane the j- complex Hermitian-orthogonal basis (l,*c). A point with the complex coordinates {z\ = X\ + iyi,z2 = x2 + 13/2) in this basis is the quaternion q = x\ + y\j + x- i>c + y2ycj.In the affine coordinate w = zijz\ on the projective line C P 1 = S? the projec-t ion 7 , of the circle in C 2 defined above is given by the formula w = tani, and thisproves Lemma 4.

We now prove Lemmas 1 and 3. In the case of Lemma 1 the direction of motionof the point of application on S? is at all times the projection of the vector field i.Consequently, the direction of motion positively co-orients the moving contact ele-ment , remaining orthogonal to the direction of the motion. Meanwhile the point ofapplication describes a great circle with speed 2 by Lemma 4.In the case of Lemma 3 the direction of motion of the point of application on S'jis at all times the projection of the vector field k. Consequently, the direction ofmotion is at all times tangent to the moving contact element. Meanwhile the point

of application describes a great circle with speed 2 by Lemma 4.Lemma 5. The complex j- structure on the sphere S^ = CP 1 defines on it theorientation in which the frame ( , , i, ,, k) ispositive.Proof of Lemma 5. Consider the vectors i and k of the fields i and k at the point 1of the sphere S3. We compute their projections on the sphere S? in the affinecoordinate w = z2/ z\ = + iy on the sphere Sj, constructed as in the proof ofLemma 4, in which we take as i.

The projections of the vectors i and k are d/ dx and d/dy, respectively (at thepoint w = 0). In fact, by the formulae of the proof of Lemma 4 with Z\ = 1, z2 = wwe find that q = 1 + xi + yk. On varying the real parameters and y near 0 wehavedq/ 8x\00 = = i(l), dq/ dy\Q0 = k = k(l),

as was asserted. Lemma 5 has been proved.Lemma 3 is proved at the same time, since the direction TTJ.k of the motion of thepoint of application is obtained from the direction , . i that co- orients the element

by turning through an angle / 2 towards the side defined by the given j- complexstructure.Lemma 2 of 7 follows from Lemma 2 of 6 and Lemma 5, since the directionof turning 'from i to k', obtained in 6, is opposite to the direction of turning ofth e given j- complex structure.


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20 V. I. Arnol'd

8. The action of right shifts on left- invariant fieldsI t turns out th at left- invariant contact structures on the sphere S 3 are mapped

i n t o each other by multiplication by appropriate quaternions on the right.We denote by R q : S 3 - S 3 multiplication on the right by th e quaternion qoflength 1.

Theorem. Multiplication of S 3 on the right by e7* maps the field j to itself whilethe fields i and k are turned through an angle 2i in the plane generated by them(from i towards k ) :

(R eit) j = i cos 2i + k sin 2t.Proof. The vectors of the fields i an d k are orthogonal t o the direction of the field j .U n d e r the isometries R q : S 3 - > S 3,q = ejt, preserving the direction j of the fibres oft h e bundle , : S 3 >S?, t he tran slated vectors remain orthogonal to the directionsof th e fibres.

The projection to S? of a vector of th e field i transported by th e isometry Rejtdoes not change under change of th e parameter t , while th e point of applicationmoves along t h e fibre (in th e direction j) with speed 1.

The projection to S? of a vector of the field i, applied at this moving point, turnswith angular velocity 2 from ^ towards Tr^k, by Lemma 2 of 7.Consequently, a vector transported by the shifts Rejt turns with angular veloc-ity + 2 relative t o i in th e direction towards k.Corollary 1. Multiplication of S 3 on the right bye71^/ 4 turns the contact i- structureof S 3 into the k- structure (and the k- structure into the i- structure).

The vector field i then becomes t h e vector field k, while k becomes i.Corollary 2. Multiplication of S 3 on the right by e71''/ 4 turns the bundle * :5 3 > Sf into the bundle ^ : S 3 > S% and T I> into . The induced maps of the2- spheres Sf - 5 | and Si - Sf are isometries. The first of these respects theorientations of these spheres given by the complex i- and k- structures, respectively,while the second changes them. Their compositions are the antipodal involutions ofthe spheres Sf and S%.

9. The duality of j- fronts and fc- fronts of i- Legendrian curvesLet L be an i- Legendrian curve (tha t is, an integral curve of the i- structure

o n S 3 ) .Proposition 1. Multiplication of S 3 by elt on the right sends an i- Legendriancurve into an i- Legendrian curve.Proof. This multiplication is an isometry of S 3. I t preserves the field i and thereforet h e contact structure orthogonal to it . Therefore integral curves of th e i- structureare mapped to integral curves.


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The geometry of spherical curves and the algebra of quaternions 21Proposition 2. Thefronts of the i- Legendrian curves L and L elt on the sphereS'j are equidistant (at a distance 2t in the direction from TTJL towards TTJ(L elt)).

Proof. This is Lemma 2 of 7.Proposition 3 . Cusps of the j- front TTJL of the i- Legendrian curve L on 5? cor-respond to points of inflection to the k- front n^L on Si and, conversely, the pointsof inflection on Sj correspond to cusps on Sf.Proof. At a cusp L has th e J- direction, th at is, it is tangent to {z e 3'}. Under th e/ c- projection th e circle {z e J i } projects to the great circle on Sf with it s normalc o n t a c t elements. Tangency of the first order (between L an d {z e J t }) in th e spaceS T * S % is transformed after projection to Sf into tangency of th e second order ofthe projections, which implies a point of inflection of th e A;- front.Proposition 4. T he front TTJ(L elt) of the shifted i- Legendrian curve L eltis congruent to the front TT^L of the original i- Legendrian curve, where = j cos 2t k sin It.Proof. Multiplication on the right by elt rotates the fields j an d k in their plane withangular velocity 2 from k towards j (by th eTheorem of 8) with cyclic permutationof th e units. The field j is sent by such a rotation into th e field .Theorem. The k- front of an i- Legendrian curve on Si is congruent to the '2- equidistant of its j- front on S j.Proof. Setting t = / 4 in Proposition 4 we obtain th e congruence of TTJ(L e"lA)an d T C k L . By Proposition 2, TTJ(L e l7 r/ 4) is th e / 2- equidistant of the front T T J L .Proof of the triality theorem from 6. The assertion concerning th e dual front hasjust been proved. Th e assertion concerning the derivative is obtained from thatproof by cyclic permutation of th e imaginary units i, j andk.

In fact, multiplication of 5 3 on the right by ekt for t = / 4 takes th e field j tothe field i, and th e field i to th e field j. Consequently it takes th e projection ^to t h e projection ,. Th e induced isometry Sf S j takes th e curve - KiL to thecurve TTJ(L e*71"/ 4). The latter curve on the sphere S j is th e derivative of the front = - KjL (by Lemma 3 of 7). The theorem is proved.Remark. I t is no t unprofitable to remark that the isometry Si > S'j constructedin th e proof (induced by multiplication on the right by e t7 r/ 4) respects th e complexorientations of the spheres Sf and Sj, while the isometry Sf > Sj (induced bymultiplication on the right by ekn/ 4) doesnot .


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22 V. I. ArnoFdC H A P T E R I I I

Q U A T E R N I O N S AND C U R V A T U R E SHere the algebra of quaternions is used to obtain additional information aboutspherical curves. First of all we consider the calculation of the geodesic curvatures

of curves on the standard two- dimensional sphere. These results are used belowin the proof of a generalized Gauss- Bonnet theorem and formulae for the Maslovindex.

10. The spherical radii of curvature of frontsConsider the front = T T J L on the sphere S? of an i- Legendrian curve L C S3.T h e centre of spherical curvatureof the front at one of its points is the point of

intersection of infinitesimally close great circles orthogonal to the front at the point.There are two such centres and they are antipodal.Definition. The spherical radiusof curvature of the front at a point is the distancefrom this point to the centre of spherical curvature along the normal that co-orientsth e front.

This radius is defined modulo .Theorem. / / an i- Legendrian curve L in S3 has at a givenpoint thedirection

x = j cos + k sin ,then its j- front KjL has at the corresponding point of the sphere 5? the sphericalradius of curvature m od .Proof The centre of curvature of the front at the given point lies on the caustic ofth e family of its equidistants: it is a singular point of the corresponding equidistant,lying on the normal to the front at that point.On multiplication of the i- Legendrian curve L on the right by eli its j- front ist u rned into its 2i- equidistant (Proposition 2 of 9). This equidistant is singularwhen the i-Legendrian curve L elt is tangent to the directions j of the fibres ofth e bundle .,.

A vector , orthogonal to i, is turned by multiplication on the right by elt inth e moving plane of the fields j and k through an angle 2t in the direction from ktowards j (by the Theorem of 8). Therefore the direction of the curve L elt (atth e given point of L) becomes tangent to the fibre ( = 0 mod ) after turning thevector through an angle mod .F or such a choice of t(= / 2) the front of the curve L elt is mod - equidistantfrom the original front T T J L . Consequently, the distance to the centre of curvatureof the front along its co- orienting normal is equal to mod .

T h e theorem just proved allows us to consider the radii of curvature of the j -fronts of i-Legendrian curves, without worrying about their smoothness.Definition. The reduced radiusof spherical curvature of a front at one of its pointsis the minimum of the absolute values of its radii of spherical curvature, min | + 0 |.This radius does not exceed / 2.


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The geometry of spherical curves and the algebra of quatern ions 23Example. The reduced radius of spherical curvature of a parallel of latitude isequal to the distance to the nearest pole along a meridian.

F r o m the theorems proved there follow:Corollary 1 . The sum of the reduced radii of spherical curvature at correspondingpoints of a front and its dual front is equal to / 2.Corollary 2. The points of spherical inflection of fronts equidistant from the givenone lie on the derivative curve of the caustic.

I n fact, at a point of the caustic the radius of curvature of the correspondingfront is equal to 0, while at a poin t of inflection of th e front it is equal to / 2.Moreover the caustic touches the ray (orthogonal to the front) at a cusp of thef ron t , lying on the caustic.

T h e theorems proved above enable one to relax the regularity of fronts andcaustics in these elementary assertions, requiring only that the i- Legendrian curveis immersed in S3.

11. Quaternions and causticsC o n s i d e r an i- Legendrian curve L on S 3. To it there corresponds the j- front

KjL on the sphere S2. The equidistants of this front are the projections of thei- Legendrian curves L eli.

C o n s i d e r th e caust ic of th is family of equidistant s. Th e family of i- Legendriancurves Lt = L elt defines a torus immersed in S3:

T2 = {l- eu :l L, i l m od 2 }.Definition. T h e caustic of the j- projection of an i- Legendrian curve is the set ofcritical values of the restriction : 2 > S2 of th e projection , : S3 S2 to thet o r u s T 2 .Theorem. The caustic of the j-projection of an i- Legendrian curve is thej-projection of a k- Legendrian curve.Proof. Fibre the torus T2 by i- Legendrian curves L t with fixed t. Such ani- Legendrian curve has at th e point I elt the j- direction for some t(l), to be precisefor t(l) = (1)/2 mod , by the Theorem of 10.Definition. The curve {I etl : I L} is the j-critical curve of the i-Legendriancurve L.Lemma 1. The j-critical curve of an i-Legendrian curve L is the set of criticalpoints of the projection : 2 5 | .Proof of Lemma 1. The tangent plane of the torus at each point is generated bya vector of the field i and a vector which is orthogonal to it and tangent to thecurve L t. This plane contains th e j- direct ion if and only if the curve L t is tangentt o it .

T h e surprising resemblance of one- dimensional caustics to one- dimensional wavefronts is clarified by the following lemma.


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24 V. I. Arnol'd

Lemma 2 (basic). The j- critical curve of an i- Legendrian curve is k- Legendrian.Proof of Lemma 2. Consider an i-Legendrian curve, having the j- direction at agiven point. We introduce a local parameter s on this curve, reducing to 0 at thispo in t . A point of the curve corresponding to a small value of s has the form

l(s) = + asz j + o(s), .T h e angle of the tangent vector of this i- Legendrian curve with the j- direction isgiven by the formula

0 ( s ) = 6s + o(s),since for s = 0 this angle is zero.

Consequently, the corresponding point of the j- crit ical curve has the forml(s) e i 9 ( s) / 2 = (z + asz j + o(s)) (1 + ibs/ 2 + o(s))

= + s(az j + (b/ 2)z ) + o(s).T h e vector in the brackets that is tangent to the j- critical curve at the point isa real linear combination of the vectors j and i of the fields j and i at thepo in t z. It is orthogonal to the vector of the field k, as the lemma asserts.

T h e theorem is proved at the same time.Thus a caustic and front on the sphere 5? are the j- projections of fc- Legendrianan d i-Legendrian curves. Therefore the symmetry between the imaginary units

allows us to extend to caustics what we already know about fronts. 12. The geodesic curvature of the derivative curve

Here we relate the geodesic curvature of the derivative curve to the derivative ofth e geodesic curvature of the original curve.Theorem 1. The geodesic curvature of the originalcurve and the anglea of itsderivative curve with the great circle tangent to the originalcurve are connected bytherelation

= tan a.Proof.Consider an infinitely thin spherical triangle, formed by two infinitely closearcs of great circles that are tangent to the original curve (at points ds apart) andof length / 2, and the derivative curve. The angles of this triangle adjacent toon e of the circular arcs are equal to and a (Fig. 8). The altitude dropped tot h a t side is equal to (tana)


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T he geometry of spherical curves and thealgebra of quaternions '

ds.

25

Figure 8

Figure 9Corollary 1. The derivative curve of a front is orthogonal to the great circle tan-gent to the front if and only if this circle is tangent to the front at a cusp (F ig. 9).Corollary 2. The derivative curve of a front is tangent to the great circle tangentto the front if and only if this circle is tangent at a point of inflection (F ig. 10).

Figure 10B o t h these corollaries allow one to estimate from below th e n u m b e r of cusps an d

points of inflection of the front (in terms of the increase of the angle a along th ederivative curve).


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26 V. I. Arnol'd

Theorem 2. The great circles touching the originalfront form a framing on itsderivative curve that is parallel with respectto the standardRiemannian metric onthesphere.Proof. Consider two points of the original front at a small distance s from eachother. We transport in parallel the tangent to the great circle touching the originalfront at the first point, from a point / of this circle on the derivative curve of thefront along the derivative curve to a point II of the circle touching the originalcurve at the second point (Fig. 11).

m

IFigure 11

We replace the transfer along the segment (/ , II) of the derivative curve bytransfer along a path (/, / / / , II) along the trajectory orthogonal to the family ofgreat circles touching the original front up to a point / / / on the second tangentcircle, and tnen along this circle touching the front to the point / / on the derivativecurve.

The area of the triangle (/, / / , / / / ) has order s2 and so the result of transfer alongthe broken path (/, / / / , II) differs from the result of transfer along the originalcurve by O(s2).

The path (/, III) has geodesic curvature 0 at the point / , since the distancefrom the point / to the centre of curvature of the path is equal to / 2. At a pointof this path at a distance from / the distance to the centre of curvature is equalto / 2 + 0( ). Therefore the geodesic curvature at this point is O(x). The distance(/, / / / ) is O(s). Therefore the total angle of turn of a vector carried along the path(/, / / / ) with respect to the directions of the normals to the path (/, / / / ) is O(s2).The path (/ / , / / / ) is geodesic, and under parallel transport of a vector along thispath the angle between the vector and the direction of the path does not change.

Finally, the direction of the great circle touching the front carried from thepoint / to the point / / along the path (/, III, / / ) is transformed to the directionof the circle touching the front at the point / / , turned through an angle O(s2).Under transfer along the derivative curve one obtains a result differing from thisby another O(s2). Therefore the derivative of the angle between the transportedvector and the directions of the great circles is equal to zero, and that means thatthe framing is parallel.Corollary. The framing of the curve on the sphere S2, obtained by j - projectionfrom the k- vectors along a j- Legendrian curve, is parallel with respect to theRiemannian metric of the sphere S2.


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T h e geometry of spherical curves and the algebra of quaternions 27I n fact, we apply Theorem 2 to the front on S 2 of an z- Legendrian curve L on

S 3/ 1. The derivative curve of th e front is the fr,- projection of th e image V ofthe curve L under the action of multiplication on the right by ekn/ 4. The framingby the directions of the great circles is the framing by the projections ., of vectorsof the field k along the curve L'. The curve V is j- Legendrian if L is i- Legendrian,since multiplication on the right by efc7r/ 4 maps the field i to the field j .

Thus by Theorem 2 for such a j- Legendrian curve V its framing by the k- vectorsprojects onto Sj as a parallel framing.

Any j- Legendrian curve may be obtained from an appropriate i- Legendrian curveby multiplication on the right by e fc7r/ 4. On applying Theorem 2 to the front ofth is i- Legehdrian curve we have th e corollary.Remark. The quatern ion units may be permuted. For example, along the projectionof an i- Legendrian curve in Sf the projections of the j- vectors (and also of thek- vectors) are parallel.

Theorem 2 enables one to compute explicitly the geodesic curvature of th e deriv-ative curve. In fact, this curvature, by Theorem 2, is equal to the derivative of theangle a between the derivative curve and the direction of the framing great circlewith respect to the natural parameter / (arc length) of the derivative curve.Theorem 3. The geodesiccurvature = da/ dl of the derivative curve is expressedin terms of the geodesic curvature >c{s) of the original curve by the formula

/ ds(VI +

which can also be written in the form = cos3 a / ds = dsina/ ds.

Corollary. The points of inflection (K = 0) of the derivative curve correspond tothe vertices ( / ds = 0) of the originalcurve.Proof of Theorem 3. Consider once again the infinitely thin spherical triangle (withsides / 2, ( / 2) ds and dl and the angle a). From this triangle we find that thelength dl of the segment of the derivative curve is ds/ cos a. But from Theorem 1we get

dx 2da r- = cos a d>c.1 +

These relations give for da/dl the expression stated in the theorem.Example 1. For a parallel of latitude = const and its derivative curve is theequator (K = 0).Example 2. Near an ordinary (semicubical) cusp the square of the radius of cur-vature of the front is proportional to the distance to the cusp.

In this case >t(s) ~ / /*; / ds ~ l / ( ^ ) 3 . The value of remains boundedas s > 0, as should be the case for an immersed curve. Therefore the is positivefor positive cusps and negative for negative cusps.


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28 V. I. Arnol'd

Example 3. Near a degenerate cusp of a front (x ~ pa, y ~ pa+1) the curvature has order si 1"0) / 0. In this case >t(s) has order s(~2)/ ~ p~ 2 (p is a regularparameter along the derivative curve).Corollary. Thegeodesic curvature of the derivative curve of a front has a zerooforder 1 at a point corresponding to a cusp of thefront of multiplicity .

H e r e = a 1: at an ord i n a ry cusp (of multiplici ty 1 ) = 2 , = 1 an d t h egeodesic c u r v a t u r e of the derivative curve at theorigin is not zero.We conclude t h a t , at a cusp of the original front of multiplici ty , vert icesof the front vanish . For = 2 after a small pe r t u r b a t io n the front has a pair ofcusps, and thevanish ing ver tex lies halfway between t h e m .T h e formulae obta ined lead to new general iza t ions of th e four -ver tex t h e o r e m(for spherical curves) , but I do not e labora te on t h i s here.

13. The derivative of a small curve andthe derivative of curvature of the curve

T h e formulae of the preceding section can be used for the study of plane curvesalso.

If the original curve is concentrated around the pole on the sphere, then itsderivative is concentrated near the equator. The geodesic curvature of a smallcurve on the sphere placed near the pole is asymptotically equal to the curvatureof the plane curve obtained by projecting it to the tangent plane at the pole.T h e curvature of the derivative curve which is close to the equator also admits asimple asymptotic expression. By combining these two formulae we obtain a ratherremarkable expression for the derivative of the radius of curvature of a plane curve,given by the following known theorem (see for example the text- book by Santalo).

Consider a curve on the Euclidean plane with Cartesian coordinates (x ,y ) ,th e parameter on which is the azimuth of the normal equator (so that(cost^)cia; + (sin )dy ~ 0 along the curve). We denote the radius of curvatureof the curve by R and we let A = sin y cos (this measures the distance fromth e origin to the normal to the curve). We shall denote thederivatives with respectto the parameter by dashes.Theorem.

R ' = A + A".Remark. From this formula there follows at once the four-vertex theorem for aconvex plane curve, namely: R' has no fewer that fourzeros.In fact A is the derivative by of the supporting function = cos +y sin of the curve, so that

R1 = - ( ' +B'").A function of the form B' + B'" has no fewer than four zeros on the circle by thetheorems of Sturm (Kellogg and Tabachnikov...), asserting that the Fourier series

F(


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T h e geometry of spherical curves and the algebra of quatern ions 29has no fewer zeros t han the number 2m of zeros of the first of its harmonics(see [2], [3], [4], [5] for details).Proof of the theorem. The formula to be proved is a limiting case of Theorem 3of 12. Let be a small parameter. Consider the small curve on the Euclideanplane, tangent to the unit sphere at the Nor t h pole. Its curvature is / , where = \/ R is the curvature of . The natural parameter on is es, where s is then a t u r a l parameter on .

Project to the sphere (for example, along the axis through the pole, or fromth e centre, or from the South pole). On the sphere we obtain a small curve con-cent ra ted near the Nor t h pole. The geodesic curvature of this curve is asymptot-ically equivalent to / as > 0, while the natural parameter sE is asymptoticallyequivalent to es. Consequently the expression of Theorem 2 of 12 for the geodesiccurvature of the derivative of

takes the form

dxejdse ~2 / dR/ds dR/ds dR= ~ = = W ~3 3 / ds Consider now the derivative of the small curve (Fig. 12). As 4 0 thisderivative curve tends to the equator. It smoothly depends on . Let us introduceon the sphere the coordinates of longitude (x = costp,y = sin^) and latitude

( = sin ).

Figure 12

Lemma 1. The derivativeofTE for small is given locallyby theequation = ( , ) , \ ( ,0)=0.

In fact, consider the great circle tangent to at the point corrresponding on toth e point whose normal has the azimuth . This circle intersects the equator at thepoin t with longitude + ( /2) + ( ) . The distance along the circle from thepoin t of tangency to the equator is equal to ( / 2) + { ) + ( 2) . The expressionfor now follows from the implicit function theorem, since the circle constructeddiffers from the meridian of longtitude + ( / 2) by a quantity of order .


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30 V. I. Arnol'd

Lemma 2. The geodesiccurvatureof asmooth curve close totheequator\ = ( , )= ( )+ ( )

is given by the formula = ( + )+ ( ).

Proof.It issufficient, forexample, toapply the Gauss- Bonnet formula to a smallcurvilinear quadrilateral bounded bythe equator, two nearby meridians and ourcurve close to the equator.T he area ofthe quadrilateral isproportional toM, while the difference oftheangles between ourcurve and the meridians is proportional to (in a firstapproximation with respect to ). From this one obtains the required expressionfor the geodesic curvature of the only side of the quadrilateral that isnot ageodesic.

Comparison ofLemmas 1and 2leads atonce to anasymptotic expression forth e geodesic curvature ofthe derivative ofthe curve at the point correspondingto the point on where the normal has the azimuth : ( ) = [ ( )+ ( )}+ ( ).

Comparing this formula with formula (1) we obtain (for the appropriate choiceof sign ofthe geodesic curvature)

dR / = A+ ,as had to beproved.Remark. The resemblance ofthe passage from the small curve to its derivative usedhere with the contactomorphism

DIM > J \O , IK)

of the space ofco-oriented contact elements ofEuclidean space to the space ofthe1-jets offunctions on the sphere used in[4] deserves special study.

CHAPTER IV

THE CHARACTERISTIC CHAIN ANDSPHERICAL INDICES OF A HYPERSURFACE

T h e index of a point with respect to a closed plane curve is not conformallyinvariant. It changes ifone takes forthe point at infinity some other point oftheRiemann sphere.I t turns out that one can adjust the indices ofall points (deducting from thema suitable number) sothat the adjusted indices are conformally invariant.In this way, with each component of the complement to acurve on the sphere onecan associate itsspherical index. The components ofthe complement, furnished


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T h e geometry of spherical curves and the algebra of quaternions 31with this index as a multiplicity, form a 2-chain, whose boundary is the originalcurve.

This remarkable 2-chain is called the characteristic chain of the curve. In an anal-ogous manner one can define the characteristic - chain of an oriented(n l)- dimensional hypersurface on an even-dimensional sphere and, moregenerally, of a wave front on an even-dimensional sphere.

Integrals over the characteristic chain appear in the Gauss- Bonnet formula forimmersed curves, in the formula for the Maslov index, and so on. Unfortunately thesimplicity and naturalness of the constructions of this and the following chapters issomewhat dimmed by the need to pay due heed and attention to endless signs andorientations.

14. The characteristic 2- chainConsider a smooth immersion of an oriented circle in general position on the

oriented 2- sphere. The characteristic 2-chain of such an immersion is the chainconsisting of the closures of the regions of the complement to the image withspecial (integral or semi- integral) coefficients. The curve itself is the boundaryof its characteristic chain. The coefficients are chosen in the following way.

Select an auxiliary point on the sphere outside the immersed curve. Weregard it as the 'point at infinity' and identify the complement of this point withth e oriented plane. The immersed curve on the sphere transforms to an immersedcurve in the plane. This curve has a well-defined index (the number of completet u r n s of the normal as one travels right round the curve). We denote this by % B (itdepends on the choice of B).

F or each point A of the plane outside the curve so obtained there is defined thewinding number of the curve around A. Since this number depends not only on Abut also on the choice of B, we denote it by i^. The number i^ depends on thecomponents of the complement to the curve in which A and lie (while is dependson the component of B).Theorem 1. The difference

i{A)=iB

A- {iB/ 2)is a conformally invariant function on the complement to the curve on thesphere:it does not depend on the choiceof the point at infinity.Example 1. Consider the equator on the oriented sphere. We orient it as theboundary of the Northern hemisphere.

This means that the frame (the outward normal of the Northern hemisphere, thevector orienting the equator) positively orients the sphere.T h e choice of orientation and co- orientation of a hypersurface in an orientedmanifold is said to be correct if on placing after the co-orienting normal an orientingframe one orients the manifold positively.In the Northern hemisphere i(A) = 1/2, while in the Southern hemispherei{A) = - 1/ 2 (Fig. 13, see p. 32). A correctly co-orienting vector is directed toth e side of smaller values of i.


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3 2 V. I. Arnol'd

Figure 13Example 2. Consider a figure of eight on the sphere (with correct orientation andco-or ienta t ion) . The figure of eight divides the sphere into three regions (Fig. 14).O n e of these (R) is bounded by both loops, the second region (P) is bounded byth e loop with co- orientation directed into the region, while the third region (Q) isbounded by the loop with co- orientation directed outside the loop. Calculationsshow that i(P) = - 1 , ~i{Q) = 1, ~ i ( R ) = 0.

Figure 14

Definition. The characteristicchain of an immersed curve in general position onth e sphere is the 2- chain consisting of the closures of the regions of the complementwith the coefficients i(A).

Suppose that we choose the point to lie in the same component of the com-plement of the immersed curve in which A lies. Then i\ = 0. This proves thefollowing corollary.Corollary. The followingformula holdsfor the characteristic chain:


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T h e geometry of spherical curves and the algebra of quaternions 33Although this formula appears to be simpler, in practice it is easier to do calcu-

lations using the formula of Theorem 1, since in it the point at infinity is fixed.Let us consider now the lift of the immersion S1 > S2 to the Legendrian curve of

th e normals formed from the correctly co-oriented contact elements of the sphere S2(tangent to the immersed curve):7 : S 1 -> ST*S2 KP3 w S3 / 1.

I remark that the bundle : ST*S2 > S2 doubly covers the Hopf bundleS3 - > S2.' T h e curve of normals' "/(S 1) lies in RP3 and is doubly covered in S 3 by a curve 7.T h e covering curve consists of one or two components, depending on the parity ofth e index of the corresponding curve on the plane. If the index % B is even, as forth e figure of eight, then there are two components, while if it is odd, then there ison e (the parity of the index does not depend on the choice of B, it is opposite tothe parity of the number of double points on the immersed curve).Theorem 2. The linking coefficientof the Legendrian curve7 in S 3 covering thecurve of normals, with the fibre of the Hopf bundle lying over the pointA e S 2\ f (S 1) , is equal to

2i{A) = 2i%- iB,independently of the parity of the index ( mod 2).

We denote by F the fibre of the bundle ST*S2 -> S2 (with its naturalor ienta t ion) .Theorem 3. Consider the 2- chain in ST*S2 whose boundary is the curve ofnormals 7 if the index of the original curve is even, and the curve 7 F if it isodd.

The projection * of the chain on S2 is the characteristic 2- chain i ofTheorem 1 (independently of the chioce of ) for even index (ig mod 2), whilep ta = i + (1/ 2) for odd index.

T h e proofs of Theorems 1-3 are given in 18 in the more general situation ofwave fronts on even- dimensional spheres. But since some facts that we requireremain true also for spheres of any dimension, I begin with these more generalfacts.

15. The indices of hypersurfaces on a sphereConsider an oriented smoothly immersed hypersurface M"""1 on an oriented

sphere Sn.Definition. The correctco-orientation of the hypersurface is the field of unitnormals such that the frame (normal, tangent frame positively orienting M)positively orients the sphere.Remark. If is the boundary of a region (with its usual orientation), then thenorma l is the outward normal of this region.


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34 V. I. Arnol'd

Consider two points A and on the sphere outside M. On taking to be thepoin t at infinity and A to be the origin we may identify SN\B with W 1 and A withO e l " . The image of in ffin will be denoted by Mf. The space Rn and thehypersurface M

Bimmersed in it are oriented.

Definition. The index ( ) of the hypersurface on the sphere relative to thepair of points (A, B) is the degree of the map of the hypersurface M% to the sphere5"1 by the rays from 0:

iBA{M) = deg(^ : Mf -> S""1) , **(*) = x/\x\( the sphere Sn~l is oriented here as the boundary of the ball |x| < 1 in W 1).Theorem 4. The index of a smoothly immersed hypersurfaceon the spherewithrespect to a pair of points of the complement depends only on the componentsofthe complement to which these points belong. The function i% on the sphere of apair of regionsof the complement to the hypersurface is skew- symmetric,and maybe represented in the form

= { )- { ),where is a function of the regionof the complement.

Consider an - chain c on 5", consisting of the closures of regions of the comple-ment of a hypersurface with coefficientsc(A)=iB,

where is an arbitrary fixed point of the complement. The sphere and the com-ponen t s here are oriented.Theorem 5. The boundary of the - dimensional chain c is the original orientedhypersurface:

dc = Mn~1.Consider a point outside the image of the hypersurface on the sphere.

Taking to infinity we identify Sn\B with W 1, as before.Definition. The index IB{M) of the immersed hypersurface with respect to thepoint is the degree of the Gauss map

iB(M) = deg(sB : ""1 - S""1) ,where SB (X ) is the normal vector of length 1 at a; which co-orients the hypersurfaceM"" 1 correctly.Theorem 6. On an even-dimensional spherethe indicesof a hypersurface relativeto points and pairs are connected by therelation

1iBA =ie - -I n particular, the parity of all the indices%A is the same.


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T h e geometry of spherical curves and the algebra of quaternions 35Theorem 7. 6 On an odd- dimensionalsphere the indices of ahypersurfacerelative to all points of its complement are the same and equal to half the Eulercharacteristicof the immersed hypersurface:

iB(M)=X(M)/ 2.T h e proofs of Theorems 1-7 will be given below, in 18.

16. Indices as linking coefficientsConsider the manifold E2n~x = S T * S n of co-oriented contact elements of the

sphere. As with any contact manifold defined by a global contact form a, it has acanonical co- orientation (da) n~ 1.

[An orienting frame can be obtained, for example, by adjoining to a frame ori-enting the base Sn a frame orienting the fibre Sn~1. The orientation of the fibremay be obtained from the orientation of the base as the orientation of the bound-ary of a small disk of radius r. The outward normals of the disk form the fibre, asr > 0. Here, as usual, the cotangent vectors are identified with the tangent vectorsby means of a metric]

Thus in E2n~l one can consider the indices of intersection of transversal chainsof complementary dimension (and also of cycles of complementary dimension).In particular, we can intersect Legendrian (n l)- dimensional chains with - dimensional ones.Definition. The Legendrian manifold of normals Ln~1 of an immersed orientedhypersurface Mn~1 in the oriented sphere 5 " is the manifold of correctly co-orientedcontact elements of the sphere tangent to the immersed hyperpsurface Mn~1.

If the original immersed manifold Mn~l is in general position in Sn, then theLegendrian submanifold U1"1 of normals is embedded in E2""1.T h e submanifold Ln~1 of normals is homologous to the fibre F"~x over an arbi-trary point b Sn\Mn~1 with some coefficient (since the cyclic group i?n__i(E) isgenerated by [F]): there is an - chain such that

da = L- kFb, keZ.

[H ere we use the homology groups of E2n~

l:if is even, then the group is J J % in dimension 1 and 0 in dimensionn;

if is odd, then it is in dimensions 1 and (and, of course, always indimensions 0 and In 1);the Ii2- homology groupsare 7L i in dimensions 0, 1, and 2n 1.]

Theorem 8. The index of an immersedhypersurface relativeto a pair ofpointsis equalto the differencebetween the indices of intersection of the chain with thefibres over these points, to be precise

The non- uniquenessof the choiceof has no effect on thedifference.Now suppose that the dimension of the ambient sphere Sn is even.6 On th is matt er see also the work of Tabachnikov [15] and the results of White discussed thereon Legendrian links in a contact manifold of dimension 4n + 1.


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36 V. I. Arnol'd

Theorem 9. On an even- dimensional sphere the index of intersection of thechain with the fibre FA is equal to

aHFA = - iA/ 2 + e/ 2,where = 0 if the index IA (and therefore also any index IB) is even, while = 1if the index IA is odd.I n particular, the projection of the - chain on the sphereSn doesnotdependfor even on the choice of , and the boundary of this projection is theoriginaloriented hypersurfaceM""1 .Remark. Theorem 9 makes clear why the indices

1(A) =iBA- (iB/2) = - iA/ 2can be interpreted as linking coefficients. In fact, for a hypersurface of even indexon an even- dimensional sphere under arbitrary choice of the point

iA =anFA-Since in this case da = L, Theorem 2 means that IA is the linking coefficientofthe Legendrian manifoldL of normals of with the fibre FA over the point A (thislinking coefficient is defined as the index of intersection of the chain filling out Lwith this fibre).T h e proofs of Theorems 8 and 9 are carried out in 18 with the help of thefollowing constructions.

17. The indices of hypersurfaces on a sphere as intersection indicesConsider two distinct points A and of the sphere Sn. For this pair of points

we construct the - chain in the manifold E2n~1 of co-oriented contact elements ofth e sphere (to intersect the Legendrian submanifolds with it ).Definition 1. The cylinder VA over the segment (A,B) is the - dimensional sub-manifold (with boundary) in E2n~1 of all contact elements of the sphere applied atth e points of the segment {A,B) smoothly embedded in the sphere.

By the segment (A, B) we understand here any smooth embedding of a segmentin the sphere joining A to (for example, a geodesic segment).T h e cylinder V^ carries a natural orientation.Definition 2. The orientation of the cylinder Vj is counted as positive if its

intersection index with the Legendrian sphere formed from the outward normals ofa small disk on 5 n with centre at A is equal to +1.Theorem 10. The index of an immersed oriented hypersurface Mn~1 with respectto a pair of points A and of the oriented sphereSn is equal to the intersectionindex of the Legendrian manifold L of normals of the immersed hypersurface with the cylinder over a segment (A,B):

Remark. Here it does not matter whether the hypersurface is correctly oriented,provided only that it is the projection on Sn of the Legendrian manifold L. In fact,changing the orientation of changes also the orientation of L and therefore alsoth e signs on both the left- and the right- hand sides of the equality.


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T h e geometry of spherical curves and the algebra of quatern ions 37Theorem 11. T heboundary of the cylinder V^ is equal to the differenceof thecoherentlyoriented fibresover the points A and B, to be precise

dVf = (- l)n(FA- FB).Consider a point on S n. For this point we construct the - chain in the manifold

E2n~l of co-oriented elements of the sphere (to intersect Legendrian manifoldswith it).Definition 3. A B- section VB is the - chain in obtained from any constantvector field on K under stereographic projection with centre B, Mn > S n\B.

Remark. The stereographic projection sends a constant field on K" to a manifolddiffeomorphic to an open ball ofdimension n. Choosing an appropriate diffeomor-phism we get a smooth map which extends to the closed ball. On the boundary wethen obtain a map of the sphere 5"1 C R n to the sphere FB (the fibre over B),given by the folding'map

> e2( , ) , (*)where e is a constant vector of length 1.

T h e folding map (*) sends any vector of the unit sphere S1"^1 to the reflectionof the constant basis vector e in the mirror orthogonal to .

O n e can easily prove the following lemma.Lemma. The foldingmap(*) is realizable geometrically as thestereographic pro-jection of a spherefrom its centre onto another sphere of the same radius passingthrough thiscentre.

Example. For = 2 the folding map (*) of a circle on a circle is the ordinarydouble covering ' -> 2 ' (Fig.15).


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38 V I. Arnol'd

T h e map of a closed ball to 2 "" 1 described above defines an - chain VB- Itsorientat ion is induced from the orientation of the base W1 5" \ B.Theorem 12. The index of a hypersurfaceMn~1 with respect to on Sn is equalto the index of intersection of the Legendrian manifold Ln~l of normals of theimmersed hypersurface Mn~l with the B- section VB'

Remark. Here it does not matter whether the mainfold is correctly oriented,provided only that it is the projection of a Legendrian manifold L to the sphere S".F or a change in the orientation of implies a change in the orientation of L andchange in the signs on both the left- and the right- hand sides of the equality.Theorem 13. The boundaryof the B- sectionVB is the image of the sphere S""1by the folding map (*), to beprecise (taking accountof the orientations)

I n other words, on even- dimensional spheres dVB = 2FB, while on odd-dimensional spheresdVs = 0.T h e geometrical Theorems 10- 13 are verifiable directly, from the definitions. For

example in Theorems 10 and 12 the contributions of each point of intersection inth e left- and in the right- hand side of the equality is the same. It is only necessaryto follow accurately through all the orientations. Theorems 1-9 are derived belowfrom these four geometrical theorems.

18. Proofs of the index theoremsProof of Theorem 6. Let be even. From Theorems 11 and 13 we have

d(2Vf -VB+ VA) = 0.

Therefore from Theorems 10 and 12 it follows that

t h a t is, Theorem 6.Proof of Theorem 4. From Theorem 11 it follows (for arbitrary n) that

d(v + v) = o, d{v + v + v) = o.Applying Theorem 10 we conclude that

iB iB iAlA - l c - %c-

By this Theorem 4 is proved.


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The geometry of spherical curves and thealgebra of quaternions 39

Proof of Theorem 5. For a fixed point the index i^(M) can becomputed as theflow of the radial field in i " S" \ with centre at the point A. The change ofthis flow corresponding to transversal positive intersection of the hypersurface by the point A isequal to 1. This proves Theorem 5.

Suppose now that the dimension ofthe sphere Sn is even.Proof of Theorem 7. Consider as an immersed hypersurface ingeneral positionin R" Sn \ B. Construct twoB- sections, corresponding to theconstant fieldsey ande\ in Er a. On calculating the degree of the Gaussian map over the points e i g Sn~1 weobtain

ie = mg - mX V m + _ x,iB

= TUQ- mf h m~_1;

where m^ is thenumber ofpoints where the co-oriented normal has the direction e i and forwhich the restriction ofthe coordinate function xihas aMorse criticalpoint of index k.

The restriction of thefunction X\ to hastherefore in all m* = m + mjjTcritical points ofindex k. Thus, = mom\ + ... +m n _ i = 2 , and Theorem 7is proved.Proof of Theorem 8. For - chains and r in E2n~x

{da) = ( - 1) ( 9 ) .Let da =L-kFc, =Vf. Then ( ) = iBAbyTheorem 10, since FcnVf = 0(the segment (A, B) may be chosen so as not to pass through C). On theotherhand,

(8 ) = ( - 1) ( FA - ) FB),according toTheorem 11. Finally,

= (3 ) ) = ( -1 ) (


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40 V. I. Arnol'd

Proof of Theorem 1.' This is a particular case of Theorem 6.Proof of Theorem 2. We use Theorem 9. Denote by the inverse image of thechain under the double covering S3 -> S T * S 2. The fibre of the Hopf bundle overA is the double covering FA of the fibre FA of the bundle S T * S 2 - S2 over A. Wehave da = 7 - eFc- Denote by DQ a 2- chain in S3 with boundary dD = Fc- Thenth e 2-chain eDc has boundary 7. Therefore the linking coefficient sought for is

f(7, FA) = ( - eDc) nFA=dnFA- eDc FA.According to Theorem 9, FA = 2 FA = - The linking coefficient ofth e fibres Fc and FA of the Hopf bundle is equal to 1, and therefore DQ FA 1 Thus 1(J,FA) = - But, by Theorem 6, - %A = 2i\ - is- And so Theorem 2 isproved.

19. The indices of fronts of Legendriansubmanifolds on an even- dimensional sphere

Let Ln~x be an oriented Legendrian immersed submanifold of the contact man-ifold S T * S n of co-oriented elements of the oriented sphere Sn.

Consider the front of L (its projection to the sphere) as an oriented andco-oriented (singular) hypersurface of the sphere.Example. Let L be the manifold of normals of an oriented hypersurface immersedin the sphere. Then for the front we

spherical curvature

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