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Aspects of Differential Geometry I [electronic resource] / by Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo.

By: Gilkey, Peter [author.].
Contributor(s): Park, JeongHyeong [author.] | Vázquez-Lorenzo, Ramón [author.] | SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Synthesis Lectures on Mathematics & Statistics: Publisher: Cham : Springer International Publishing : Imprint: Springer, 2015Edition: 1st ed. 2015.Description: XIII, 140 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783031024078.Subject(s): Mathematics | Statistics  | Engineering mathematics | Mathematics | Statistics | Engineering MathematicsAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 510 Online resources: Click here to access online
Contents:
Preface -- Acknowledgments -- Basic Notions and Concepts -- Manifolds -- Riemannian and Pseudo-Riemannian Geometry -- Bibliography -- Authors' Biographies -- Index .
In: Springer Nature eBookSummary: 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. Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index.
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Preface -- Acknowledgments -- Basic Notions and Concepts -- Manifolds -- Riemannian and Pseudo-Riemannian Geometry -- Bibliography -- Authors' Biographies -- Index .

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. Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index.

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