INTRODUCTION TO DIFFERENTIABLE MANIFOLDS
CONTENTS CHAPTER I DIFFERENTIAL CALCULUS §1 CATEGORIES §2 FINITE DIMENSIONAL VECTOR SPACES §3 DERIVATIVES AND COMPOSITION OF MAPS §4 INTEGRATION AND TAYLOR’S FORMULA §5 THE INVERSE MAPPING THEOREM CHAPTER II MANIFOLDS §1 ATLASES, CHARTS, MORPHISMS §2 SUBMANIFOLDS, IMMERSIONS, SUBMERSIONS §3 PARTITIONS OF UNITY §4 MANIFOLDS WITH BOUNDARY CHAPTER III VECTOR BUNDLES §1 DE?NITION, PULL BACKS §2 THE TANGENT BUNDLE §3 EXACT SEQUENCES OF BUNDLES §4 OPERATIONS ON VECTOR BUNDLES §5 SPLITTING OF VECTOR BUNDLES CHAPTER IV VECTOR FIELDS AND DIFFERENTIAL EQUATIONS §1 EXISTENCE THEOREM FOR DI¤ERENTIAL EQUATIONS §2 VECTOR FIELDS, CURVES, AND FLOWS §3 SPRAYS §4 THE FLOW OF A SPRAY AND THE EXPONENTIAL MAP §5 EXISTENCE OF TUBULAR NEIGHBORHOODS §6 UNIQUENESS OF TUBULAR NEIGHBORHOODS CHAPTER V OPERATIONS ON VECTOR FIELDS AND DIFFERENTIAL FORMS §1 VECTOR FIELDS, DI¤ERENTIAL OPERATORS, BRACKETS §2 LIE DERIVATIVE §3 EXTERIOR DERIVATIVE §4 THE POINCARE´ LEMMA §5 CONTRACTIONS AND LIE DERIVATIVE §6 VECTOR FIELDS AND 1-FORMS UNDER SELF DUALITY §7 THE CANONICAL 2-FORM §8 DARBOUX’S THEOREM CHAPTER VI THE THEOREM OF FROBENIUS §1 STATEMENT OF THE THEOREM §2 DI¤ERENTIAL EQUATIONS DEPENDING ON A PARAMETER §3 PROOF OF THE THEOREM §4 THE GLOBAL FORMULATION §5 LIE GROUPS AND SUBGROUPS CHAPTER VII METRICS §1 DE?NITION AND FUNCTORIALITY §2 THE METRIC GROUP §3 REDUCTION TO THE METRIC GROUP §4 METRIC TUBULAR NEIGHBORHOODS §5 THE MORSE LEMMA §6 THE RIEMANNIAN DISTANCE §7 THE CANONICAL SPRAY CHAPTER VIII INTEGRATION OF DIFFERENTIAL FORMS §1 SETS OF MEASURE 0 §2 CHANGE OF VARIABLES FORMULA §3 ORIENTATION §4 THE MEASURE ASSOCIATED WITH A DI¤ERENTIAL FORM CHAPTER IX STOKES’ THEOREM §1 STOKES’ THEOREM FOR A RECTANGULAR SIMPLEX §2 STOKES’ THEOREM ON A MANIFOLD §3 STOKES’ THEOREM WITH SINGULARITIES CHAPTER X APPLICATIONS OF STOKES’ THEOREM §1 THE MAXIMAL DE RHAM COHOMOLOGY §2 VOLUME FORMS AND THE DIVERGENCE §3 THE DIVERGENCE THEOREM §4 CAUCHY’S THEOREM §5 THE RESIDUE THEOREM