EAN: 9783031012792

SYNTHESIS LECTURES ON MATHEMATICS & STATISTICS ...

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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