Shoshichi Kobayashi was born January 4, 1932 in Kofu, Japan. After obtaining his mathematics degree from the University of Tokyo and his Ph.D. from the University of Washington, Seattle, he held positions at the Institute for Advanced Study, Princeton, at MIT and at the University of British Columbia between 1956 and 1962, and then moved to the University of California, Berkeley, where he is now Professor in the Graduate School.Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with K. Nomizu, Hyperbolic Complex Manifolds and Holomorphic Mappings and Differential Geometry of Complex Vector Bundles.

• VOLUME IInterdependence of the Chapters and the Sections xi  Chapter IDifferentiable Manifolds1. Differentiable manifolds 12. Tensor algebras 173. Tensor fields 264. Lie groups 385. Fibre bundles 50  Chapter IITheory of Connections1. Connections in a principle fibre bundle 632. Existence and extension of connections 673. Parallelism 684. Holonomy groups 715. Curvature for and structure equation 756. Mappings of connections 797. Reduction theorem 838. Holonomy theorem 899. Flat connections 9210. Local and infinitesimal holonomy groups 9411. Invariant connections 103  Chapter IIILinear and Affine Connections1. Connections in a vector bundle 1132. Linear connections 1183. Affine connections 1254. Developments 1305. Curvature and torsion tensors 1326. Geodesics 1387. Expressions in local coordinate systems 1408. Normal coordinates 1469. Linear infitesimal holonomy groups 151  Chapter IVRiemannian Connections1. Riemannian metrics 1542. Riemannian connections 1583. Normal coordinates and convex neighborhoods 1624. Completeness 1725. Holonomy groups 1796. The decomposition theorem of de Rham 1877. Affine holonomy groups  Chapter VCurvature and Space Forms1. Algebraic preliminaries 1982. Sectional curvature3. Spaces of constant curvature 2044. Flat affine and Riemannian connections 209  Chapter VITransformations1. Affine mappings and affine transformations 2252. Infinitesimal affine transformations 2293. Isometries and infinitesimal isometries 2364. Holonomy and infinitesimal isometries 2445. Ricci tensor and infinitesimal isometries 2486. Extension of local isomorphisms 2527. Equivalence problem 256  Appendices1. Ordinary linear differential equations 2672. A connected, locally compact metric space is separable 2693. Partition of unity 2724. On an arcwise connected subgroup of a Lie group 2755. Irreducible subgroups of O(n) 2776. Green's theorem 2817. Factorization lemma 284  Notes1. Connections and holonomy groups 2872. Complete affine and Riemannian connections 2913. Ricci tensor and scalar curvature 2924. Spaces of constant positive curvature 2945. Flat Riemannian manifolds 2976. Parallel displacement of curvature 3007. Symmetric spaces 3008. Linear connections with recurrent curvature 3049. The automorphism group of a geometric structure 30610. Groups of isometries and affine transformations with maximum dimensions 30811. Conformal transformations of a Riemannian manifold 309  Summary of Basic Notations 313Bibliography 315  Index 325  Errata for Foundations of Differential Geometry, Volume I 330Errata for Foundations of Differential Geometry, Volume II 331VOLUME II  Chapter VIISubmanifolds1. Frame bundles of a submanifold 12. The Gauss map 63. Covariant differentiation and second fundamental form 104. Equations of Gauss and Codazzi 225. Hypersurfaces in a Euclidean space 296. Type number and rigidity 427. Fundamental theorem for hypersurfaces 478. Auto-parallel submanifolds and totally geodesic submanifolds 53  Chapter VIIIVariations of the Length Integral1. Jacobi fields 632. Jacobi fields in a Rimannian manifold 683. Conjugate points 714. Comparison theorem 765. The first and second variations of the length integral 796. Index theorem of Morse 887. Cut loci 968. Spaces of non-positive curvature 1029. Center of gravity and fixed points of isometries 108  Chapter IXComplex Manifolds1. Algebraic preliminaries 1142. Almost complex manifolds and complex manifolds 1213. Connections in almost complex manifolds 1414. Hermitian metrics and Kaehler metrics 1465. Kaehler metrics in local coordinate systems 1556. Examples of Kaehler manifolds 1597. Holomorphic sectional curvature 1658. De Rham decomposition of Kaehler manifolds 1719. Curvature of Kaehler submanifolds 17510. Hermitian connections in Hermitian vector bundles 178  Chapter XHomogeneous Spaces1. Invariant affine connections 1862. Invariant connections on reductive homogeneous spaces 1903. Invariant indefinite Riemannian metrics 2004. Holonomy groups of invariant connections 2045. The de Rham decomposition and irreducibility 2106. Invariant almost complex structures 216  Chapter XISymmetric Spaces1. Affine locally symmetric spaces 2222. Symmetric spaces 2253. The canonical connection on symmetric space 2304. Totally geodesic submanifolds 2345. Structure of symmetric Lie algebras 2386. Riemannian symmetric spaces 2437. Structure of orthogonal symmetric Lie algebras 2468. Duality 2539. Hermitian symmetric spaces 25910. Examples 26411. An outline of the classification theory  Chapter XIICharacteristic Classes1. Weil homomorphism 2932. Invaraint polynomials 2983. Chern classes 3054. Pontrjagin classes 3125. Euler classes 314  Appendices8. Integrable real analytic almost complex structures 3219. Some definitions and facts on Lie algebras 325  Notes12. Connections and holonomy groups (Supplement to Note 1) 33113. The automorphism group of geometric structure (Supplement to Note 9) 33214. The Laplacian 33715. Surafces of constant curvature in R3 34316. Index of nullity 34717. Type number and rigidity of imbedding 34918. Isometric imbeddings 35419. Equivalence problems for Riemannian manifolds 35720. Gauss-Bonnet theorem 35821. Total curvature 36122. Topology of Riemannian manifolds with positive curvature 36423. Topology of Kaehler manifolds with positive curvature 36824. Structure theorems on homogeneous complex manifols 37325. Invariant connections on homogeneous spaces 37526. Complex submanifolds 37827. Minimal submanifolds 37928. Contact structure and related structures 381  Bibliography 387  Summary of Basic Notations 455  Index for Volumes I and II 459  Errata for Foundations of Differential Geometry, Volume I 469Errata for Foundations of Differential Geometry, Volume II 470

Shoshichi Kobayashi was born January 4, 1932 in Kofu, Japan. After obtaining his mathematics degree from the University of Tokyo and his Ph.D. from the University of Washington, Seattle, he held positions at the Institute for Advanced Study, Princeton, at MIT and at the University of British Columbia between 1956 and 1962, and then moved to the University of California, Berkeley, where he is now Professor in the Graduate School.Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with K. Nomizu, Hyperbolic Complex Manifolds and Holomorphic Mappings and Differential Geometry of Complex Vector Bundles.

• VOLUME IInterdependence of the Chapters and the Sections xi  Chapter IDifferentiable Manifolds1. Differentiable manifolds 12. Tensor algebras 173. Tensor fields 264. Lie groups 385. Fibre bundles 50  Chapter IITheory of Connections1. Connections in a principle fibre bundle 632. Existence and extension of connections 673. Parallelism 684. Holonomy groups 715. Curvature for and structure equation 756. Mappings of connections 797. Reduction theorem 838. Holonomy theorem 899. Flat connections 9210. Local and infinitesimal holonomy groups 9411. Invariant connections 103  Chapter IIILinear and Affine Connections1. Connections in a vector bundle 1132. Linear connections 1183. Affine connections 1254. Developments 1305. Curvature and torsion tensors 1326. Geodesics 1387. Expressions in local coordinate systems 1408. Normal coordinates 1469. Linear infitesimal holonomy groups 151  Chapter IVRiemannian Connections1. Riemannian metrics 1542. Riemannian connections 1583. Normal coordinates and convex neighborhoods 1624. Completeness 1725. Holonomy groups 1796. The decomposition theorem of de Rham 1877. Affine holonomy groups  Chapter VCurvature and Space Forms1. Algebraic preliminaries 1982. Sectional curvature3. Spaces of constant curvature 2044. Flat affine and Riemannian connections 209  Chapter VITransformations1. Affine mappings and affine transformations 2252. Infinitesimal affine transformations 2293. Isometries and infinitesimal isometries 2364. Holonomy and infinitesimal isometries 2445. Ricci tensor and infinitesimal isometries 2486. Extension of local isomorphisms 2527. Equivalence problem 256  Appendices1. Ordinary linear differential equations 2672. A connected, locally compact metric space is separable 2693. Partition of unity 2724. On an arcwise connected subgroup of a Lie group 2755. Irreducible subgroups of O(n) 2776. Green's theorem 2817. Factorization lemma 284  Notes1. Connections and holonomy groups 2872. Complete affine and Riemannian connections 2913. Ricci tensor and scalar curvature 2924. Spaces of constant positive curvature 2945. Flat Riemannian manifolds 2976. Parallel displacement of curvature 3007. Symmetric spaces 3008. Linear connections with recurrent curvature 3049. The automorphism group of a geometric structure 30610. Groups of isometries and affine transformations with maximum dimensions 30811. Conformal transformations of a Riemannian manifold 309  Summary of Basic Notations 313Bibliography 315  Index 325  Errata for Foundations of Differential Geometry, Volume I 330Errata for Foundations of Differential Geometry, Volume II 331VOLUME II  Chapter VIISubmanifolds1. Frame bundles of a submanifold 12. The Gauss map 63. Covariant differentiation and second fundamental form 104. Equations of Gauss and Codazzi 225. Hypersurfaces in a Euclidean space 296. Type number and rigidity 427. Fundamental theorem for hypersurfaces 478. Auto-parallel submanifolds and totally geodesic submanifolds 53  Chapter VIIIVariations of the Length Integral1. Jacobi fields 632. Jacobi fields in a Rimannian manifold 683. Conjugate points 714. Comparison theorem 765. The first and second variations of the length integral 796. Index theorem of Morse 887. Cut loci 968. Spaces of non-positive curvature 1029. Center of gravity and fixed points of isometries 108  Chapter IXComplex Manifolds1. Algebraic preliminaries 1142. Almost complex manifolds and complex manifolds 1213. Connections in almost complex manifolds 1414. Hermitian metrics and Kaehler metrics 1465. Kaehler metrics in local coordinate systems 1556. Examples of Kaehler manifolds 1597. Holomorphic sectional curvature 1658. De Rham decomposition of Kaehler manifolds 1719. Curvature of Kaehler submanifolds 17510. Hermitian connections in Hermitian vector bundles 178  Chapter XHomogeneous Spaces1. Invariant affine connections 1862. Invariant connections on reductive homogeneous spaces 1903. Invariant indefinite Riemannian metrics 2004. Holonomy groups of invariant connections 2045. The de Rham decomposition and irreducibility 2106. Invariant almost complex structures 216  Chapter XISymmetric Spaces1. Affine locally symmetric spaces 2222. Symmetric spaces 2253. The canonical connection on symmetric space 2304. Totally geodesic submanifolds 2345. Structure of symmetric Lie algebras 2386. Riemannian symmetric spaces 2437. Structure of orthogonal symmetric Lie algebras 2468. Duality 2539. Hermitian symmetric spaces 25910. Examples 26411. An outline of the classification theory  Chapter XIICharacteristic Classes1. Weil homomorphism 2932. Invaraint polynomials 2983. Chern classes 3054. Pontrjagin classes 3125. Euler classes 314  Appendices8. Integrable real analytic almost complex structures 3219. Some definitions and facts on Lie algebras 325  Notes12. Connections and holonomy groups (Supplement to Note 1) 33113. The automorphism group of geometric structure (Supplement to Note 9) 33214. The Laplacian 33715. Surafces of constant curvature in R3 34316. Index of nullity 34717. Type number and rigidity of imbedding 34918. Isometric imbeddings 35419. Equivalence problems for Riemannian manifolds 35720. Gauss-Bonnet theorem 35821. Total curvature 36122. Topology of Riemannian manifolds with positive curvature 36423. Topology of Kaehler manifolds with positive curvature 36824. Structure theorems on homogeneous complex manifols 37325. Invariant connections on homogeneous spaces 37526. Complex submanifolds 37827. Minimal submanifolds 37928. Contact structure and related structures 381  Bibliography 387  Summary of Basic Notations 455  Index for Volumes I and II 459  Errata for Foundations of Differential Geometry, Volume I 469Errata for Foundations of Differential Geometry, Volume II 470