Differential Geometry

PREFACE
CONTENTS
1 DIFFERENTIABLE MANIFOLDS
1 Differentiable Manifolds
1.1 Topological spaces and introductory material
1.2 Differentiable manifolds
1.3 Functions on manifolds
2 Vector Fields
2.1 Tangent vectors
2.2 Vector fields
2.3 Differentials of mappings
2.4 Submanifolds
3 Tensor Fields
3.1 Tensor algebra
3.2 Tensor fields
3.3 Lie derivation and exterior differentiation
3.4 Distributions
4 Lie Groups and Lie Algebras (Part I)
4.1 Lie groups and Lie algebras
4.2 GL(n;R) and GL(n;C)
4.3 Lie transformation groups
2 THEORY OF CONNECTIONS
1 Fibre Bundles
1.1 Principal fibre bundles and associated fibre bundles
1.2 Reduction of structure groups
1.3 Covering manifolds
2 Connections in Principal Fibre Bundles
2.1 Connection in P(M,G)
2.2 Parallel displacement and holonomy groups
2.3 Curvature form and structure equations
2.4 Homomorphisms of connections
2.5 Holonomy theorem
2.6 Local holonomy groups
2.7 Infinitesimal holonomy groups
2.8 Flat connections
2.9 Connections in associated bundles
3 Linear Connections
3.1 Canonical 1-forms and connection forms
3.2 Covariant differentiation
3.3 Expression of canonical 1-forms and connections forms in local coordinates
3.4 Expression of covariant derivatives in local coordinates
3.5 Bianchi identities
3.6 Linear holonomy groups
3.7 Affine connections
3.8 Development
3.9 Geodesics
3.10 Exponential mappings
3.11 Normal coordinates
3.12 Projective connections
3.13 Connections in vector bundles
3 RIEMANNIAN MANIFOLDS
1 Riemann Metrics and Riemann Connections
1.1 Bundles of orthonormal frames
1.2 Riemann connections
2 Metric Properties of Riemannian Manifolds
2.1 Expression of Riemann connections on orthonormal frames
2.2 Structure equations in polar systems
2.3 Completeness
3 Sectional Curvature ans Spaces of Constant Curvature
3.1 Sectional curvature of a Riemannian manifold
3.2 Isometry and sectional curvature
3.3 Schur's theorem
3.4 Model spaces of spaces of constant curvature
3.5 Conformal transformations
4 Holonomy Groups of Riemannian Manifolds
4.1 Holonomy groups of pseudo-Riemannian manifolds
4.2 Decomposition of holonomy groups
4.3 Affine holonomy groups
5 Curvature Tensor-Preserving Transformations
5.1 Curvature transformations and invariant distributions
5.2 Curvature tensor-preserving transformations
4 THEORY OF SUBMANIFOLDS
1 Bundles of Frames over Riemannian Submanifolds
1.1 Bundles of orthonormal frames over Riemannian submanifolds and normal bundles
1.2 Expression of induced connections in terms of orthonormal frames
2 Covariant Derivatives in Riemannian Submanifolds
2.1 Connecting quantities
2.2 Equattions of Gauss and Weingarten
2.3 Gauss, Codazzi, and Ricci equations
2.4 Absolute, relative, and normal curvatures
3 Isometric Immersions in Euclidean Spaces
3.1 Riemannian submanifolds of Euclidean spaces
3.2 Isometric embedding
3.3 Minimal immersions
4 The Gauss Map
4.1 Gauss map ψ_M → G(n,p) in R^{n+p}
4.2 Induced metric on Gauss images
4.3 Tension fields and harmonic Gauss maps
4.4 Minimal Gauss map
5 Affine Submanifolds
5.1 Induced connections
5.2 Affine normals of affine hypersurfaces in A^{n+1}
5 COMPLEX MANIFOLDS
1 Algebraic Preliminaries
1.1 Complexification and a complex structure of a real vector space
1.2 Decomposition of V^c of V(=R_J^{2n}. Hermition inner product
1.3 Functions on C^n
2 Complex Manifolds and Almost Complex Manifolds
2.1 Complex manifolds
2.2 Almost complex manifolds
2.3 Integrability of a C^ω almost complex structure
2.4 Almost complex affine connections
3 Metric Almost Complex Connections
3.1 Metric connections on almost Hermition and Hermition manifolds
3.2 Expresssion in unitary frames of Riemann connections on Kählerian submanifolds
3.3 Expression of Riemann connections on a Kählerian manifold in complex local coordinates
3.4 Kählerian manifolds of constant holomorhic sectional curvature
3.5 Kählerian submanifolds
6 HOMOGENEOUS AND SYMMETRIC SPACES
1 Lie Groups and Lie Algebras (Part II)
1.1 Complexification and real forms. Nilpotent and solvable Lie algebras
1.2 Prelimary facts on representations
1.3 Representations of solvable and nilpotnent Lie algebras
1.4 Structure of semisimple Lie algebras
1.5 Weyl bases and compact real forms
1.6 Dynkin and extended Dynkin diagrams
2 Invariant Connections in Reductive Homogeneous Spaces
2.1 Reductive homogeneous spaces
2.2 Invariant affine connections
2.3 Canonical affine connections
2.4 Affine connections invariant under parallelism
3 Symmetric Spaces
3.1 Affine symmetric spaces
3.2 Symmetric spaces
3.3 Irreducible symmetric spaces
3.4 Riemannian symmetric spaces
3.5 Structure of the orthogonal involulative Lie algebra
3.6 Sectional curvature of Riemannian symmeric spaces. Minimal immersions
3.7 Totally geodesic submanifolds
3.8 Hermitian symmetric spaces
3.9 Outline of classification of symmeric spaces
7 G-STRUCTURES AND TRANSFORMATION GROUPS
1 G-Structures
1.1 Definition of G-structures
1.2 G-structures defined by tensors
1.3 G-connections
1.4 Prolongation and type number of linear Lie algebras
1.5 Jets and frames of higher order
1.6 Möbius groups K(n)
1.7 Cartan connections in P(Ξ^n, K(n))
1.8 Conformal structure and normal conformal connections
2 Groups of Automorphism
2.1 Automorphisms of G-structures
2.2 Groups of automorphisms
3 Groups of Affine Transformations
3.1 Infinitesimal affine transformations
3.2 Integrability condition of ℒ_X ∇=0
3.3 Classification of affinely connected manifolds with torsion-free connections by the order of groups of affine transformations
4 Groups of Isometries on Riemannian Manifolds
4.1 Infinitesimal isometries
4.2 Structure of Riemannian manifolds admitting groups of isometriesof order r=(1/2)n(n-1)+1
8 CALCULUS OF VARIATIONS FOR LENGTHS OF GEODESICS
1 Synge's Formula
1.1 Jacobi equattions in affinely connected manifolds
1.2 Synge's formulas
1.3 Focal and conjugate points
1.4 The Gauss lemma
2 Comparison Theorems
2.1 The Mmorse and Rauch comparison theorem
2.2 The Morse and Schoenberg comparison theorem
3 Cut Locus and the Index Theorem
3.1 Cut locus
3.2 Closed geodesics in lens spaces
3.3 Index theorem
9 THE DE RHAM THEOREM, CHARACTERISTIC CLASSES, AND HARMONIC FORMS
1 de Rham Cohomology Theory
1.1 Integration over chains
1.2 de Rham cohomology
1.3 Sheaves and presheaves
1.4 Fine and torsion-free sheaves. Resolutions
1.5 Cochain complexes
1.6 Abstract sheaf cohomology
1.7 Classical sheaf cohomologies
1.8 Proof of the de Rham theorem
2 Characteristic Classes
2.1 The Weil mapping
2.2 The Weil theorem
2.3 Invariant polynomials. Special characteristic classes
3 Harmonic Functions and Forms
3.1 The Laplace-Beltrami operator
3.2 ∆f = nμf on compact and orientable Riemannian manifolds
3.3 Decomposition theorem and the Hodge fundamental theorem for compact and orientable Riemannian manifolds
3.4 Curvature and Betti numbers
3.5 The operators L and Λ
3.6 Harmonic forms on Kählerian manifolds
3.7 Effective forms on Kählerian manifolds
NOTATION
BIBLIOGRAPHY
INDEX