## Comparison Theorems in Riemannian GeometryComparison Theorems in Riemannian Geometry |

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### Contents

1 | |

Chapter 2 Toponogovs Theorem | 42 |

Chapter 3 Homogeneous spaces | 55 |

Chapter 4 Morse theory | 80 |

Chapter 5 Closed geodesics and the cut locus | 93 |

Chapter 6 The Sphere Theorem and its generalizations | 106 |

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acts properly discontinuously acts uniformly arc length argument assume bi-invariant metric broken geodesics Chapter Cheeger closed geodesic compact manifold complete the proof conjugate point constant curvature convex set Corollary curve deﬁned deﬁnition denote diffeomorphism dimension endpoints equation exists expp fact ﬁnd ﬁnite ﬁrst conjugate point ﬁrst variation formula ﬂat follows easily geodesic segment geometry Hence hinge homeomorphic homogeneous spaces implies inﬁnite inner product Jacobi ﬁeld l-parameter Let G Lie algebra Lie group local isometry Math Milnor minimal geodesic minimal segment neighborhood nonnegative curvature normal coordinate ball p(yo parallel ﬁeld perpendicular piecewise smooth positive curvature Proof of Theorem properly discontinuously Proposition Rauch riemannian manifold riemannian metric satisﬁes second variation sectional curvature sequence simply connected Sphere Theorem subgroup submanifold subspace suﬁiciently symmetric space tangent space tangent vector topological Toponogov’s Theorem totally geodesic triangle inequality unique minimal geodesic universal covering space vanishes vector ﬁeld zero