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" Aspects of differential geometry I / "
Peter Gilkey (Univesrity of Oregon, Eugene, OR), JeongHyeong Park (Sungkyunkwan University, Suwon, Korea, Institute for Advanced Study, Seoul, Korea), Ramón Vázquez-Lorenzo (University of Santiago de Compostela, Santiago de Compostela, Spain).
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BL
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854877
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| Main Entry
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Gilkey, Peter B.
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| Title & Author
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Aspects of differential geometry I /\ Peter Gilkey (Univesrity of Oregon, Eugene, OR), JeongHyeong Park (Sungkyunkwan University, Suwon, Korea, Institute for Advanced Study, Seoul, Korea), Ramón Vázquez-Lorenzo (University of Santiago de Compostela, Santiago de Compostela, Spain).
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| Publication Statement
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San Rafael, California :: Morgan & Claypool Publishers,, [2015]
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, ©2015
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| Series Statement
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Synthesis lectures on mathematics and statistics,; #15
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1 online resource (xiii, 140 pages) :: illustrations
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| ISBN
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1627056629
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: 1627056637
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: 1627057846
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: 9781627056625
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: 9781627056632
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: 9781627057844
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1627056629
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1627057838
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9781627056625
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9781627057837
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Includes bibliographical references (pages 131-133) and index.
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| Contents
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1. Basic notions and concepts -- 1.1 Metric spaces -- 1.2 Linear algebra -- 1.3 The derivative -- 1.4 The inverse and implicit function theorems -- 1.5 The Riemann integral -- 1.6 Improper integrals -- 1.7 The change of variable theorem -- 2. Manifolds -- 2.1 Smooth manifolds -- 2.2 The tangent and cotangent bundles -- 2.3 Stokes' theorem -- 2.4 Applications of stokes' theorem -- 3. Riemannian and pseudo-Riemannian geometry -- 3.1 The pseudo-Riemannian measure -- 3.2 Connections -- 3.3 The Levi-Civita connection -- 3.4 Geodesics -- 3.5 The Jacobi operator -- 3.6 The Gauss-Bonnet theorem -- 3.7 The Chern-Gauss-Bonnet theorem.
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| Abstract
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Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.
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| Subject
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Geometry, Differential.
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Geometry, Differential.
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MATHEMATICS-- Geometry-- General.
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| Dewey Classification
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516.36
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| LC Classification
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QA641.G552 2015
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| Added Entry
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Park, Jeonghyeong
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Vázquez-Lorenzo, Ramón
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