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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 Torrón Universidade de Santiago Angel Ferrández Izquierdo Universidad de Murcia

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A. M. Naveira, 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 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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A. M. Naveira, 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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A. M. Naveira, 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 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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A. M. Naveira, 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 Rodeja Prof. 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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A. M. Naveira, While working on his thesis, Antonio teaches Mathematics at the Faculty of Sciences, University of Santiago de Compostela, as Assistant Professor from 1965 to He obtained a position as “Profesor Adjunto de Universidad” in 1973, staying at Santiago till 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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A. M. Naveira, 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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A. M. Naveira, 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 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 Antonio

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

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Antonio M. Naveira Mathematical 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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Foliations Obstructions to the integrability of a totally geodesic distributions: If F is totally geodesic of dimension 2k+1, the the Pontryagin groups Pont k ( F ) = 0, for all k dim M. (with D.L. Jonson, Geometriae Dedicata 11 (1981), ) Riemannian foliations: (Ph. D.) A geodesic, which is orthogonal to some leave, meets orthogonally any other (Collectanea Matematica 21 (1970), 1-61) UV RpRp RpRp γ uv fufu fvfv f u, f v projections γ uv co-cycle diffeomorphisms Reeb foliation

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Almost Hermitian Manifolds (M,g,J) almost Hermitian manifold J 2 =-id,g(JX,JY)=g(X,Y), (X,Y)=g(JX,Y) Kähler manifold Complex Symplectic 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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Almost Hermitian Manifolds 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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Almost Hermitian Manifolds 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 S 6 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), ) 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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Almost Hermitian Manifolds 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), ) (With L. Vanhecke, Demonstratio Math. 10 (1977), ) (With A. Ferrández, Czech. Math. J. 32 (1983), )

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Almost Product Manifolds (M,g,P) almost product metric manifold P 2 =id,g(PX,PY) = g(X,Y) TM = V + H, pr H :TM H, pr V :TM V P = pr V - pr H Totally geodesic foliation U P=0 Integrable distribution ( U P)V=( V P)U Almost foliated distribution ( U P)U=0 D 1 -property (X)=0, (X)=2g( eu e u,X) D2-property g(( U P)V+( V P)U,X) = 2/p g(U,V) (X)

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Almost Product Manifolds A classification of almost product manifolds, Rend. Mat. Appl. 3 (1983),

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Almost Product Manifolds Angel Montesinos, Michigan Math. J. 30 (1983), Francisco Carreras, Math. Proc. Cambridge Philos. Soc. 91 (1982), Vicente Miquel, Pacific J. Math. 111 (1984), Olga Gil, Canad. Math. Bull. 26 (1983), Rend. Circ. Mat. Palermo (2) 32 (1983), Antonio Hernández Rocamora, Illinois J. Math. 32 (1988), 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

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