## Riemannian Geometry and Geometric AnalysisThe present textbook is a somewhat expanded version of the material of a three-semester course I gave in Bochum. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. In the first chapter, we introduce the basic geometric concepts, like dif ferentiable manifolds, tangent spaces, vector bundles, vector fields and one parameter groups of diffeomorphisms, Lie algebras and groups and in par ticular Riemannian metrics. We also derive some elementary results about geodesics. The second chapter introduces de Rham cohomology groups and the es sential tools from elliptic PDE for treating these groups. In later chapters, we shall encounter nonlinear versions of the methods presented here. The third chapter treats the general theory of connections and curvature. In the fourth chapter, we introduce Jacobi fields, prove the Rauch com parison theorems for Jacobi fields and apply these results to geodesics. These first four chapters treat the more elementary and basic aspects of the subject. Their results will be used in the remaining, more advanced chapters that are essentially independent of each other. In the fifth chapter, we develop Morse theory and apply it to the study of geodesics. The sixth chapter treats symmetric spaces as important examples of Rie mannian manifolds in detail. |

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### Contents

1 | |

De Rham Cohomology and Harmonic | 55 |

Parallel Transport Connections | 76 |

Geodesics and Jacobi Fields | 125 |

A Short Survey on Curvature and Topology 165 | 164 |

Morse Theory and Closed Geodesics | 173 |

Symmetric Spaces and Kähler Manifolds | 211 |

The PalaisSmale Condition and Closed Geodesics 263 | 262 |

Harmonic Maps | 277 |

Linear Elliptic Partial Differential Equations | 385 |

Fundamental Groups and Covering Spaces | 393 |

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algebra apply assume basis boundary bounded called chart choose claim compact complete compute condition conformal connection Consequently consider constant contained continuous converges coordinates Corollary critical points curvature curve defined Definition denote depends derivative determined diffeomorphism differentiable manifold dimension energy equation equivalent estimate Euclidean example exists finite follows formula function geodesic geometry given harmonic map hence holds holomorphic homotopic implies induces isometry Jacobi field Lemma length linear Math means metric minimal Namely negative neighborhood normal obtain operator orthonormal parallel particular positive Proof Remark result Riemannian manifold Riemannian metric satisfies sectional curvature sequence smooth solution sphere structure submanifold surface symmetric space tangent tangent vector tensor Theorem theory transformation unique values vanishes variation vector bundle vector field write