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TRANSCRIPT
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
On the curvatures of subalgebras of nilpotent Lie algebras
Ana Hinic GalicLa Trobe University, Australia
coauthors: Grant Cairns, Yury Nikolayevsky (La Trobe University, Australia)Marcel Nicolau (Universitat Autonoma de Barcelona, Spain)
PADGE2012, KULeuven, Belgium
August 29, 2012
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Table of contents
1 Nilpotent Lie algebras
2 Curvatures of a nilpotent Lie algebrasMetric Lie algebrasSectional curvatureRicci curvatureScalar curvature
3 Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras
Let g be an n-dimensional Lie algebra over R.
• Defined the following ideals:C0(g) = g ,C1(g) = [g, g],Ck+1(g) = [Ck(g), g], for all k ≥ 0.
Then we have the descending central series of g:
g = C0(g) ⊃ C1(g) ⊃ · · · ⊃ Ck(g) ⊃ . . . .
Definition
A Lie algebra g is called nilpotent if there is an integer k such that
Ck(g) = {0}.
The smallest integer k such that Ck(g) = {0} is called the nilindex of g.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras
Let g be an n-dimensional Lie algebra over R.
• Defined the following ideals:C0(g) = g ,C1(g) = [g, g],Ck+1(g) = [Ck(g), g], for all k ≥ 0.
Then we have the descending central series of g:
g = C0(g) ⊃ C1(g) ⊃ · · · ⊃ Ck(g) ⊃ . . . .
Definition
A Lie algebra g is called nilpotent if there is an integer k such that
Ck(g) = {0}.
The smallest integer k such that Ck(g) = {0} is called the nilindex of g.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebras
Let g be an n-dimensional Lie algebra over R.
• Defined the following ideals:C0(g) = g ,C1(g) = [g, g],Ck+1(g) = [Ck(g), g], for all k ≥ 0.
Then we have the descending central series of g:
g = C0(g) ⊃ C1(g) ⊃ · · · ⊃ Ck(g) ⊃ . . . .
Definition
A Lie algebra g is called nilpotent if there is an integer k such that
Ck(g) = {0}.
The smallest integer k such that Ck(g) = {0} is called the nilindex of g.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Examples of nilpotent Lie algebras
1 Every abelian Lie algebra is nilpotent with the nilindex equal to 1.
2 The Heisenberg algebra h2k+1 defined in the basis {X1,X2, . . . ,X2k+1} by
[X2i−1,X2i ] = X2k+1 , i = 1, . . . , k.
The nilindex is equal to 2.
3 The n-dimensional algebra m0(n) defined in a basis {X1, . . . ,Xn} by thebrackets
[X1,Xi ] = Xi+1 for all 2 ≤ i ≤ n − 1.
The nilindex is equal to n − 1.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Metric Lie algebras
• Let (G , g) be a simply-connected Lie group with left-invariant Riemannianmetric g .
• Then (g, 〈·, ·〉) is the corresponding Lie algebra of G equipped with an innerproduct (a metric Lie algebra).
• If g is a metric Lie algebra, with inner product 〈·, ·〉, the Levi-Civitaconnection on g is given by:
2〈∇XY ,Z〉 = 〈[X ,Y ],Z〉+ 〈[Z ,X ],Y 〉+ 〈[Z ,Y ],X 〉, ∀X ,Y ,Z ∈ g. (1)
• Decomposition:
∇XY =1
2[X ,Y ] + U(X ,Y ), where
〈U(X ,Y ),Z〉 =1
2(〈[Z ,X ],Y 〉+ 〈[Z ,Y ],X 〉) .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Metric Lie algebras
• Let (G , g) be a simply-connected Lie group with left-invariant Riemannianmetric g .
• Then (g, 〈·, ·〉) is the corresponding Lie algebra of G equipped with an innerproduct (a metric Lie algebra).
• If g is a metric Lie algebra, with inner product 〈·, ·〉, the Levi-Civitaconnection on g is given by:
2〈∇XY ,Z〉 = 〈[X ,Y ],Z〉+ 〈[Z ,X ],Y 〉+ 〈[Z ,Y ],X 〉, ∀X ,Y ,Z ∈ g. (1)
• Decomposition:
∇XY =1
2[X ,Y ] + U(X ,Y ), where
〈U(X ,Y ),Z〉 =1
2(〈[Z ,X ],Y 〉+ 〈[Z ,Y ],X 〉) .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Sectional curvature
• For a metric Lie algebra g, the sectional curvature for X ,Y ∈ g:
K(X ,Y ) = − R(X ,Y ,X ,Y )
〈X ,X 〉〈Y ,Y 〉 − 〈X ,Y 〉2 . (2)
• The numerator k of the curvature function K for X ,Y ∈ g is equal to
k(X ,Y ) =− R(X ,Y ,X ,Y )
=‖U(X ,Y )‖2 − 〈U(X ,X ),U(Y ,Y )〉 − 3
4‖[X ,Y ]‖2
− 1
2〈[X , [X ,Y ]],Y 〉 − 1
2〈[Y , [Y ,X ]],X 〉.
(3)
Theorem (Wolf, 1964)
Let (G , g) be a connected nonabelian nilpotent Lie group and let g be thecorresponding Lie algebra.Then there exist two-dimensional subspaces π1, π2, π3 ⊂ g such that thesectional curvatures satisfy
K(π1) < 0 < K(π3) and K(π2) = 0.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Sectional curvature
• For a metric Lie algebra g, the sectional curvature for X ,Y ∈ g:
K(X ,Y ) = − R(X ,Y ,X ,Y )
〈X ,X 〉〈Y ,Y 〉 − 〈X ,Y 〉2 . (2)
• The numerator k of the curvature function K for X ,Y ∈ g is equal to
k(X ,Y ) =− R(X ,Y ,X ,Y )
=‖U(X ,Y )‖2 − 〈U(X ,X ),U(Y ,Y )〉 − 3
4‖[X ,Y ]‖2
− 1
2〈[X , [X ,Y ]],Y 〉 − 1
2〈[Y , [Y ,X ]],X 〉.
(3)
Theorem (Wolf, 1964)
Let (G , g) be a connected nonabelian nilpotent Lie group and let g be thecorresponding Lie algebra.Then there exist two-dimensional subspaces π1, π2, π3 ⊂ g such that thesectional curvatures satisfy
K(π1) < 0 < K(π3) and K(π2) = 0.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Ricci curvature
• Ricci curvature tensor: Ric(X ,Y ) =∑n
i=1 R(Ei ,X ,Y ,Ei ) where{E1,E2, . . . ,En} is an orthonormal basis for g.
• Ricci curvature in the direction of X ∈ g (X 6= 0) is
Ric(X ) =Ric(X ,X )
‖X‖2 . (4)
• I. Dotti, 1982: For a metric Lie algebra g, the Ricci curvature function in adirection X ∈ g is given by
ric(X ) = Ric(X ,X ) =n∑
i=1
R(Ei ,X ,X ,Ei )
= −1
2
n∑i=1
‖[X ,Ei ]‖2 −1
2B(X ,X ) +
1
4
n∑i,j=1
〈X , [Ei ,Ej ]〉2 −n∑
i=1
〈[U(Ei ,Ei ),X ],X 〉,
(5)
where B(X ,Y ) = tr(ad(X ) ◦ ad(Y )) is the Killing form.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Ricci curvature
• Ricci curvature tensor: Ric(X ,Y ) =∑n
i=1 R(Ei ,X ,Y ,Ei ) where{E1,E2, . . . ,En} is an orthonormal basis for g.
• Ricci curvature in the direction of X ∈ g (X 6= 0) is
Ric(X ) =Ric(X ,X )
‖X‖2 . (4)
• I. Dotti, 1982: For a metric Lie algebra g, the Ricci curvature function in adirection X ∈ g is given by
ric(X ) = Ric(X ,X ) =n∑
i=1
R(Ei ,X ,X ,Ei )
= −1
2
n∑i=1
‖[X ,Ei ]‖2 −1
2B(X ,X ) +
1
4
n∑i,j=1
〈X , [Ei ,Ej ]〉2 −n∑
i=1
〈[U(Ei ,Ei ),X ],X 〉,
(5)
where B(X ,Y ) = tr(ad(X ) ◦ ad(Y )) is the Killing form.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Lemma
Let g be a nilpotent metric Lie algebra and {E1,E2, . . . ,En} an orthonormalbasis. Then for all X ,Y ∈ g
(a) ric(X ) = 14
∑ni,j=1〈X , [Ei ,Ej ]〉2 − 1
2
∑ni=1 ‖[X ,Ei ]‖2,
(b) Ric(X ,Y ) = 14
∑ni,j=1〈[Ei ,Ej ],X 〉〈[Ei ,Ej ],Y 〉 − 1
2
∑ni=1〈[X ,Ei ], [Y ,Ei ]〉.
Theorem (Milnor, 1976)
For any left-invariant metric on a nonabelian nilpotent Lie group there exists adirection of strictly negative Ricci curvature and a direction of strictly positiveRicci curvature.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Lemma
Let g be a nilpotent metric Lie algebra and {E1,E2, . . . ,En} an orthonormalbasis. Then for all X ,Y ∈ g
(a) ric(X ) = 14
∑ni,j=1〈X , [Ei ,Ej ]〉2 − 1
2
∑ni=1 ‖[X ,Ei ]‖2,
(b) Ric(X ,Y ) = 14
∑ni,j=1〈[Ei ,Ej ],X 〉〈[Ei ,Ej ],Y 〉 − 1
2
∑ni=1〈[X ,Ei ], [Y ,Ei ]〉.
Theorem (Milnor, 1976)
For any left-invariant metric on a nonabelian nilpotent Lie group there exists adirection of strictly negative Ricci curvature and a direction of strictly positiveRicci curvature.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Scalar curvature
The scalar curvature of a metric Lie algebra g is given by
s =n∑
i=1
Ric(Ei )
= −1
4
n∑i,j=1
‖[Ei ,Ej ]‖2 −1
2
n∑i=1
B(Ei ,Ei )− ‖n∑
i=1
U(Ei ,Ei )‖2.(6)
Lemma
Suppose that g is an n-dimensional metric Lie algebra with an orthonormalbasis {E1, . . . ,En} with respect to which the commutator coefficients are c k
i,j ;
that is, [Ei ,Ej ] =∑n
k=1 c ki,j Ek .
If g is nilpotent, then its scalar curvature is
s = −1
4
n∑i,j,k=1
(c ki,j
)2
.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Metric Lie algebrasSectional curvatureRicci curvatureScalar curvature
Scalar curvature
The scalar curvature of a metric Lie algebra g is given by
s =n∑
i=1
Ric(Ei )
= −1
4
n∑i,j=1
‖[Ei ,Ej ]‖2 −1
2
n∑i=1
B(Ei ,Ei )− ‖n∑
i=1
U(Ei ,Ei )‖2.(6)
Lemma
Suppose that g is an n-dimensional metric Lie algebra with an orthonormalbasis {E1, . . . ,En} with respect to which the commutator coefficients are c k
i,j ;
that is, [Ei ,Ej ] =∑n
k=1 c ki,j Ek .
If g is nilpotent, then its scalar curvature is
s = −1
4
n∑i,j,k=1
(c ki,j
)2
.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Consider a Lie subgroup H of a simply-connected Lie group G and thecorresponding Lie algebras h and g respectively.
• ∇h: the Levi-Civita connection on H defined by the restriction of 〈·, ·〉g to h.
• The second fundamental form of g, defined by Gauss’ formula as
α(X ,Y ) = ∇X Y −∇hXY ,
has the explicit form
α(X ,Y ) =1
2
r∑j=1
(〈[fj ,X ],Y 〉+ 〈[fj ,Y ],X 〉) fj (7)
for an orthonormal basis {f1, . . . , fr} of the orthogonal complement h⊥ of h.
• Let {h1, . . . , hm} be an orthonormal basis for a subalgebra h of g.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Consider a Lie subgroup H of a simply-connected Lie group G and thecorresponding Lie algebras h and g respectively.
• ∇h: the Levi-Civita connection on H defined by the restriction of 〈·, ·〉g to h.
• The second fundamental form of g, defined by Gauss’ formula as
α(X ,Y ) = ∇X Y −∇hXY ,
has the explicit form
α(X ,Y ) =1
2
r∑j=1
(〈[fj ,X ],Y 〉+ 〈[fj ,Y ],X 〉) fj (7)
for an orthonormal basis {f1, . . . , fr} of the orthogonal complement h⊥ of h.
• Let {h1, . . . , hm} be an orthonormal basis for a subalgebra h of g.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Let h be an ideal of a nilpotent Lie algebra g and f ∈ h⊥.
• Then [f , h] ⊂ h and ad(f ) is nilpotent, so the restriction of ad(f ) to h isnilpotent and hence
tr(ad(f ) |h) = 0.
• Then (7) impliesm∑
i=1
α(hi , hi ) = 0
so the mean curvature ‖∑m
i=1 α(hi , hi )‖ of h is equal to zero.
Lemma
Let H be a connected Lie subgroup of a simply-connected nilpotent Lie groupG endowed with a left-invariant Riemannian metric and let the correspondingLie algebras be h and g, respectively.If h is an ideal of g, then H is a minimal submanifold of G .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Let h be an ideal of a nilpotent Lie algebra g and f ∈ h⊥.
• Then [f , h] ⊂ h and ad(f ) is nilpotent, so the restriction of ad(f ) to h isnilpotent and hence
tr(ad(f ) |h) = 0.
• Then (7) impliesm∑
i=1
α(hi , hi ) = 0
so the mean curvature ‖∑m
i=1 α(hi , hi )‖ of h is equal to zero.
Lemma
Let H be a connected Lie subgroup of a simply-connected nilpotent Lie groupG endowed with a left-invariant Riemannian metric and let the correspondingLie algebras be h and g, respectively.If h is an ideal of g, then H is a minimal submanifold of G .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
• The intrinsic curvatures of h:Rh,Kh,Rich, s(h) the curvature operator, sectional curvature, Ricci curvatureand scalar curvature, respectively, defined by ∇h.
• The extrinsic curvatures of h:Rh
e ,Khe ,Rich
e , se(h) the curvature operator, sectional curvature, Ricci curvatureand scalar curvature by using the Levi-Civita connection ∇ of G .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
• The intrinsic curvatures of h:Rh,Kh,Rich, s(h) the curvature operator, sectional curvature, Ricci curvatureand scalar curvature, respectively, defined by ∇h.
• The extrinsic curvatures of h:Rh
e ,Khe ,Rich
e , se(h) the curvature operator, sectional curvature, Ricci curvatureand scalar curvature by using the Levi-Civita connection ∇ of G .
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Intrinsic and extrinsic sectional curvatures
• For X ,Y ∈ h, the extrinsic sectional curvature is just K(X ,Y ) and thecorresponding formula is given by
K(X ,Y ) = Kh(X ,Y )− 〈α(X ,X ), α(Y ,Y )〉 − ‖α(X ,Y )‖2
‖X‖2‖Y ‖2 − 〈X ,Y 〉2 ,
for linearly independent (not necessary orthonormal) vector fields X ,Y ∈ h.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
• For X ,Y ∈ h, the extrinsic Ricci curvature tensor is
Riche (X ,Y ) =
m∑i=1
R(hi ,X ,Y , hi ).
• The extrinsic Ricci curvature in a direction of a unit vector field X ∈ h is
Riche (X ) = Rich
e (X ,X ). (8)
Theorem
Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis{h1, . . . , hm} for h. If X ∈ h, then the Ricci curvature function satisfies
riche (X ) = rich(X ) +
m∑i=1
‖α(X , hi )‖2,
while for X 6= 0, the extrinsic Ricci curvature satisfies
Riche (X ) = Rich(X ) +
m∑i=1
‖α(X , hi )‖2
‖X‖2 . (9)
In particular, Riche (X ) ≥ Rich(X ).
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Corollary
If a is an abelian ideal of a nilpotent Lie algebra g, thenthe extrinsic Ricci curvature Rica
e (X ) is nonnegative for all X ∈ a.
You cannot just replace the Ricci curvature by the sectional curvature.It is not true that for an abelian ideal a, one has K(X ,Y ) ≥ 0 for all X ,Y ∈ a.Indeed, consider the five-dimensional algebra generated by the orthonormalbasis {X1, . . . ,X5} with relations
[X1,X2] = X3, [X1,X4] = X5.
X2, . . . ,X5 generate a four-dimensional abelian ideal. However, forX = (X2 + X3)/2 and Y = (X4 + X5)/2 we have
K(X ,Y ) = −1/4.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Corollary
If a is an abelian ideal of a nilpotent Lie algebra g, thenthe extrinsic Ricci curvature Rica
e (X ) is nonnegative for all X ∈ a.
You cannot just replace the Ricci curvature by the sectional curvature.It is not true that for an abelian ideal a, one has K(X ,Y ) ≥ 0 for all X ,Y ∈ a.Indeed, consider the five-dimensional algebra generated by the orthonormalbasis {X1, . . . ,X5} with relations
[X1,X2] = X3, [X1,X4] = X5.
X2, . . . ,X5 generate a four-dimensional abelian ideal. However, forX = (X2 + X3)/2 and Y = (X4 + X5)/2 we have
K(X ,Y ) = −1/4.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Theorem
Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis{h1, . . . , hm} for h. Then
se(h) = s(h) +m∑
i,j=1
‖α(hi , hj)‖2. (10)
In particular, se(h) ≥ s(h).
Corollary
If a is an abelian ideal of a nilpotent Lie algebra g, thenthe extrinsic scalar curvature se(a) of a is nonnegative.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Theorem
Let h be an ideal of a nilpotent Lie algebra g and choose an orthonormal basis{h1, . . . , hm} for h. Then
se(h) = s(h) +m∑
i,j=1
‖α(hi , hj)‖2. (10)
In particular, se(h) ≥ s(h).
Corollary
If a is an abelian ideal of a nilpotent Lie algebra g, thenthe extrinsic scalar curvature se(a) of a is nonnegative.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Corollary
If a1, a2 are abelian ideals of a nilpotent Lie algebra g, with a1 ⊂ a2, then
se(a1) ≤ se(a2).
An ideal with the maximal extrinsic scalar curvature may not be abelian.Consider the five-dimensional Lie algebra g generated by the orthonormal basis{X1, . . . ,X5} with the relations
[X1,X2] = 2X3, [X1,X3] =1
2X4, [X1,X4] = X5, [X2,X3] =
1
2X5.
Let hk be the ideal generated by Xk , . . . ,X5. So one has the seriesg = h1 ⊃ · · · ⊃ h5.
k 1 2 3 4 5
se(hk) −114
52
34
12
0
The maximum occurs at h2, which is not abelian.The algebra g in this example has the property that it has a unique maximalabelian ideal, h3.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Corollary
If X ,Y generate a two-dimensional ideal a of a nilpotent Lie algebra g, then
K(X ,Y ) ≥ 0.
You cannot just replace ideal by subalgebraConsider the five-dimensional algebra generated by the orthonormal basis{X1, . . . ,X5} with relations
[X1,X2] = X3, [X1,X4] = X5.
Then X = (X2 + X3)/2 and Y = (X4 + X5)/2 generate a two-dimensionalabelian subalgebra, while K(X ,Y ) = −1/4.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
Corollary
If X ,Y generate a two-dimensional ideal a of a nilpotent Lie algebra g, then
K(X ,Y ) ≥ 0.
You cannot just replace ideal by subalgebraConsider the five-dimensional algebra generated by the orthonormal basis{X1, . . . ,X5} with relations
[X1,X2] = X3, [X1,X4] = X5.
Then X = (X2 + X3)/2 and Y = (X4 + X5)/2 generate a two-dimensionalabelian subalgebra, while K(X ,Y ) = −1/4.
Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras
Nilpotent Lie algebrasCurvatures of a nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebras
Intrinsic and extrinsic curvatures of subalgebras of nilpotent Lie algebrasIntrinsic and extrinsic sectional curvaturesIntrinsic and extrinsic Ricci curvaturesIntrinsic and extrinsic scalar curvatures
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Ana Hinic Galic La Trobe University, Australia On the curvatures of subalgebras of nilpotent Lie algebras