INTRODUCTION TO SMOOTH MANIFOLDS
CONTENTS SMOOTH MANIFOLDS TOPOLOGICAL MANIFOLDS SMOOTH STRUCTURES EXAMPLES LOCAL COORDINATE REPRESENTATIONS MANIFOLDS WITH BOUNDARY PROBLEMS SMOOTH MAPS SMOOTH FUNCTIONS AND SMOOTH MAPS SMOOTH COVERING MAPS LIE GROUPS BUMP FUNCTIONS AND PARTITIONS OF UNITY THE TANGENT BUNDLE TANGENT VECTORS PUSH-FORWARDS COMPUTATIONS IN COORDINATES THE TANGENT SPACE TO A MANIFOLD WITH BOUNDARY TANGENT VECTORS TO CURVES ALTERNATIVE DE?NITIONS OF THE TANGENT SPACE VECTOR FIELDS THE COTANGENT BUNDLE COVECTORS TANGENT COVECTORS ON MANIFOLDS THE DI?ERENTIAL OF A FUNCTION PULLBACKS LINE INTEGRALS CONSERVATIVE COVECTOR FIELDS SUBMANIFOLDS SUBMERSIONS, IMMERSIONS, AND EMBEDDINGS EMBEDDED SUBMANIFOLDS THE INVERSE FUNCTION THEOREM AND ITS FRIENDS LEVEL SETS IMAGES OF EMBEDDINGS AND IMMERSIONS RESTRICTING MAPS TO SUBMANIFOLDS VECTOR FIELDS AND COVECTOR FIELDS ON SUBMANIFOLDS LIE SUBGROUPS EMBEDDING AND APPROXIMATION THEOREMS SETS OF MEASURE ZERO IN MANIFOLDS THE WHITNEY EMBEDDING THEOREM THE WHITNEY APPROXIMATION THEOREM LIE GROUP ACTIONS GROUP ACTIONS ON MANIFOLDS EQUIVARIANT MAPS QUOTIENTS OF MANIFOLDS BY GROUP ACTIONS COVERING MANIFOLDS QUOTIENTS OF LIE GROUPS HOMOGENEOUS SPACES TENSORS THE ALGEBRA OF TENSORS TENSORS AND TENSOR FIELDS ON MANIFOLDS SYMMETRIC TENSORS RIEMANNIAN METRICS DI?ERENTIAL FORMS THE HEURISTICS OF VOLUME MEASUREMENT THE ALGEBRA OF ALTERNATING TENSORS THE WEDGE PRODUCT DI?ERENTIAL FORMS ON MANIFOLDS EXTERIOR DERIVATIVES SYMPLECTIC FORMS 10 INTEGRATION ON MANIFOLDS ORIENTATIONS ORIENTATIONS OF HYPERSURFACES INTEGRATION OF DI?ERENTIAL FORMS STOKES’S THEOREM MANIFOLDS WITH CORNERS INTEGRATION ON RIEMANNIAN MANIFOLDS 11 DE RHAM COHOMOLOGY THE DE RHAM COHOMOLOGY GROUPS HOMOTOPY INVARIANCE COMPUTATIONS THE MAYER–VIETORIS THEOREM SINGULAR HOMOLOGY AND COHOMOLOGY THE DE RHAM THEOREM 12 INTEGRAL CURVES AND FLOWS INTEGRAL CURVES FLOWS THE FUNDAMENTAL THEOREM ON FLOWS COMPLETE VECTOR FIELDS PROOF OF THE ODE THEOREM 13 LIE DERIVATIVES THE LIE DERIVATIVE LIE BRACKETS COMMUTING VECTOR FIELDS LIE DERIVATIVES OF TENSOR FIELDS APPLICATIONS 14 INTEGRAL MANIFOLDS AND FOLIATIONS TANGENT DISTRIBUTIONS INTEGRAL MANIFOLDS AND INVOLUTIVITY THE FROBENIUS THEOREM FOLIATIONS 15 LIE ALGEBRAS AND LIE GROUPS LIE ALGEBRAS INDUCED LIE ALGEBRA HOMOMORPHISMS ONE-PARAMETER SUBGROUPS THE EXPONENTIAL MAP THE CLOSED SUBGROUP THEOREM LIE SUBALGEBRAS AND LIE SUBGROUPS THE FUNDAMENTAL CORRESPONDENCE REFERENCES