Differential geometry and topology: with a view to dynamical systems

MANIFOLDSIntroductionReview of topological conceptsSmooth manifoldsSmooth mapsTangent vectors and the tangent bundleTangent vectors as derivationsThe derivative of a smooth mapOrientationImmersions, embeddings and submersionsRegular and critical points and valuesManifolds with boundarySard's theoremTransversalityStabilityExercisesVECTOR FIELDS AND DYNAMICAL SYSTEMSIntroductionVector fieldsSmooth dynamical systemsLie derivative, Lie bracketDiscrete dynamical systemsHyperbolic fixed points and periodic orbitsExercisesRIEMANNIAN METRICSIntroductionRiemannian metricsStandard geometries on surfacesExercisesRIEMANNIAN CONNECTIONS AND GEODESICSIntroductionAffine connectionsRiemannian connectionsGeodesicsThe exponential mapMinimizing properties of geodesicsThe Riemannian distanceExercisesCURVATUREIntroductionThe curvature tensorThe second fundamental formSectional and Ricci curvaturesJacobi fieldsManifolds of constant curvatureConjugate pointsHorizontal and vertical sub-bundlesThe geodesic flowExercisesTENSORS AND DIFFERENTIAL FORMSIntroductionVector bundlesThe tubular neighborhood theoremTensor bundlesDifferential formsIntegration of differential formsStokes' theoremDe Rham cohomologySingular homologyThe de Rham theoremExercisesFIXED POINTS AND INTERSECTION NUMBERSIntroductionThe Brouwer degreeThe oriented intersection numberThe fixed point indexThe Lefschetz numberThe Euler characteristicThe Gauss-Bonnet theoremExercisesMORSE THEORYIntroductionNondegenerate critical pointsThe gradient flowThe topology of level setsManifolds represented as CW complexesMorse inequalitiesExercisesHYPERBOLIC SYSTEMSIntroductionHyperbolic setsHyperbolicity criteriaGeodesic flowsExercisesReferencesIndex