Riemannian Geometry: A Modern Introduction
This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.
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arc length argument Assume bounded bundle calculation chart comparison theorem conjugate points consider constant sectional curvature convex coordinates Corollary cut locus cut point deck transformation group define deﬁned Deﬁnition denote diffeomorphic differentiable dimensional distance domain eigenvalue equal equation Exercise exists exponential map ﬁeld ﬁnite ﬁrst follows formula function Gauss curvature given grad Hint implies the claim integral isometry isoperimetric inequality Jacobi field Lemma Levi-Civita connection mean curvature minimizing geodesic n–dimensional neighborhood normal Note oriented orthogonal orthonormal basis parallel translation path Proof Proposition Prove respect Ricci curvature Riemannian manifold Riemannian measure Riemannian metric satisfying second fundamental form sectional curvature simply connected smooth boundary sphere submanifold subset symmetric theorem Theorem unit speed geodesic valid vanishes vector field volume
Page 10 - R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math.
Page 18 - Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds. J.