DIFFERENTIAL GEOMETRY VALENCIA 2001

An International Meeting on the Occasion of the 60th. Birthday of Professor A. M. Naveira

A Tour on the Life and Work of

Antonio Martínez Naveira

Luis Hervella TorrónUniversidade de Santiago

Angel Ferrández IzquierdoUniversidad de Murcia

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• Antonio was born at the very small village of Churío (Aranga) in 1940

• He attended the elementary school at Aranga under teachers Rey de Castro and Mosquera.

A. M. Naveira: Some Biographical DataA. M. Naveira: Some Biographical Data

With the support of Rey de Castro’s family, Antonio moves to La Coruña city, where he attends the secondary school at “Academia Galicia”, and gets the Bachelor Degree with Honors in 1959

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• During this period of secondary studies, Antonio also attends the preparatory courses for School Teachers, getting his habilitation in 1959

It is specially difficult for him to pass the caligraphy exam ...

... those who know him can easily understand the reason.

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• In 1960 Antonio passes the Entrance Examination to the University. He studies Mathematics at the Faculty of Sciences of the University of Santiago de Compostela, where he graduated in 1965.

Antonio receiving the diploma acknowledging his scholarship from Dr. A. J. Echeverri,

President of the University.

During this period, Antonio lives at “Colegio Mayor San Clemente”

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1969

• After his graduation, Antonio follows graduated studies under Prof. E. Vidal Abascal.

He will become his Ph. D. Thesis Advisor, with a strong influence in Antonio’s career

Antonio presenting his Thesis, evaluated by the Committee:

Prof. García RodejaProf. Etayo, Prof. Vidal Abascal,Prof. Vaquer, and Prof. Viviente

In January 1969, Antonio presents his thesis “Variedades foliadas con métrica casi-fibrada”, obtaining the maximal calification

• In 1973 Antonio defended his Ph. D. Thesis at Paris: “Quelques propriétés du tenseur de courbure des variétés kaeheleriennes et presque-kaehleriennes. Leurs applications au Lemma de Schur” under Professors. Lichnerowicz, Dolbeault and Deheuvels

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• While working on his thesis, Antonio teaches Mathematics at the Faculty of Sciences, University of Santiago de Compostela, as Assistant Professor from 1965 to 1973.

• He obtained a position as “Profesor Adjunto de Universidad” in 1973, staying at Santiago till 1975.

• In 1975 Antonio moves to Granada when he obtains a position as “Profesor Agregado de Geometría V (Diferencial)” at the University of Granada.

• Later, he moves to Valencia in 1976 as “Catedrático de Geometría V (Diferencial)” at University of Valencia where he teaches Mathematics at the present.

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• My personal relation with Antonio comes from 1966, when his future wife, Isabel Vázquez Paredes and me studied Mathematics at Santiago.

At that time he was teaching the second year of Calculus ......

Next year they married and since then their home became my favorite restaurant (!!).

In 1974 I presented my Doctoral Thesis, under the advise of Prof. E. Vidal and A. M. Naveira.

I have had the privilege of being, not only the first student, but also a friend of Antonio.

With Antonio and Tata in Paris, 1973

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• Work capacity: Even nowadays it is difficult for his students to work as hard as he uses to do.

• Optimisim: It is amazing the way he attacks all kind of problems, not only the mathematical ones.

Antonio was the president of the Royal Spanish Mathematical Society during the period 1996-2000. The Society was completely dead before him. Now we all are very optimistic about the future of the R.S.M.E., whose honorary President is Prince Felipe.

About AntonioAbout Antonio

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Durham, 1974“Congress on Riemannian Geometry”We met prof. Alfred Gray and Lieven Vanhecke

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Mathematical activity of prof. NaveiraMathematical activity of prof. Naveira

1. Foliations

2. Almost Hermitian manifolds

3. Almost product manifolds

4. Volumes of small geodesic spheres and tubes

5. Integral Geometry

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• Obstructions to the integrability of a totally geodesic distributions:

If F is totally geodesic of dimension 2k+1, the the Pontryagin groupsPontk(F) = 0, for all k dim M.

(with D.L. Jonson, Geometriae Dedicata 11 (1981), 347-352)

• Riemannian foliations: (Ph. D.)

A geodesic, which is orthogonal to some leave, meets orthogonally any other

(Collectanea Matematica 21 (1970), 1-61)

U V

Rp Rpγuv

fu fv

fu, fv projections

γuv co-cycle diffeomorphisms

Reeb foliation

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(M,g,J) almost Hermitian manifold

J2=-id, g(JX,JY)=g(X,Y), (X,Y)=g(JX,Y)

Kähler manifold ComplexSymplectic J=0

The holomorphic sectional curvature of a Kähler manifold is constant at a point mεM if an only if the curvature tensor at m satisfies

Consequences:• The function c(m) is constant provided that M is connected.• A Kähler manifold of constant holomorphic sectional curvature is locally

- a complex projective space- a complex Euclidean space- a complex hyperbolic space

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In a natural way, there exists sixteen classes of almost Hermitian manifolds.

Problem 1. Does it exist a Schur-like lemma for the holomorphic sectional curvature of some more general classes of almost Hermitian manifolds ?

Problem 2. Does it exist a local classification theorem for almost Hermitian manifolds of constant holomorphic sectional curvature ?

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Consequences:

• There exist a Schur lemma for the constancy of the holomorphic sectional curvature of Nearly- Kähler manifolds.

• A Nearly Kähler manifold of constant holomorphic sectional curvature is locally- a complex space form- the six-dimensional sphere S6 with the structure induced from the Cayley numbers.

In the special case of being (M,g,J) a Nearly-Kähler manifold, R* reduces to (with L. Hervella, Proc. Amer. Math. Soc. 49 (1974), 421-425)

The holomorphic sectional curvature of an almost Hermitian manifold is pointwise constant if and only if

where R* is the curvature tensor

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• Higher order curvature operators

• Chern numbers and formally holomorphic connections

• Curvature identities for almost Hermitian manifolds

• Normal forms of curvature operators

(With M. Barros, A. Gray, L. Vanhecke, J. Reine Angew. Math. 314 (1980), 89-98)

(J. Differential Geom. 9 (1974), 55-60)

(With M. Barros, C. R. Acad. Sci. Paris 284 (1977), 1461-1463)(With L. Vanhecke, Demonstratio Math. 10 (1977), 189-203)

(With A. Ferrández, Czech. Math. J. 32 (1983), 358-364)

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(M,g,P) almost product metric manifold

P2=id, g(PX,PY) = g(X,Y)

TM = V+H, prH:TM H, prV:TM VP = prV - prH

Totally geodesic foliation UP=0

Integrable distribution (UP)V=(VP)U

Almost foliated distribution (UP)U=0

D1-property (X)=0, (X)=2g( eueu,X)

D2-property g((UP)V+(VP)U,X) = 2/p g(U,V) (X)

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A classification of almost product manifolds, Rend. Mat. Appl. 3 (1983), 577-592.

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Angel Montesinos, Michigan Math. J. 30 (1983), 31-36.

Francisco Carreras, Math. Proc. Cambridge Philos. Soc. 91 (1982), 99-106.

Vicente Miquel, Pacific J. Math. 111 (1984), 163-178.

Olga Gil, Canad. Math. Bull. 26 (1983), 358-364.Rend. Circ. Mat. Palermo (2) 32 (1983), 315-329.

Antonio Hernández Rocamora, Illinois J. Math. 32 (1988), 654-671.

• Irreducibility of the subspaces in Naveira’s classification

• Examples of different kinds of almost product metric manifolds

• Examples of the different classes in Naveira’s classification

• Geometrical properties of the different classes of almost product metric manifolds• Curvature-relations: topological obstructions

• Harmonic and weak-harmonic distributions