Differential geometry is a subject related to many fields in mathematics and the sciences. The authors of this book provide a vertically integrated introduction to differential geometry and geometric analysis. The material is presented in three distinct parts: an introduction to geometry via submanifolds of Euclidean space, a first course in Riemannian geometry, and a graduate special topics course in geometric analysis, and it contains more than enough content to serve as a good textbook for a course in any of these three topics.
The reader will learn about the classical theory of submanifolds, smooth manifolds, Riemannian comparison geometry, bundles, connections, and curvature, the Chern-Gauss-Bonnet formula, harmonic functions, eigenfunctions, and eigenvalues on Riemannian manifolds, minimal surfaces, the curve shortening flow, and the Ricci flow on surfaces. This will provide a pathway to further topics in geometric analysis such as Ricci flow, used by Hamilton and Perelman to solve the Poincare and Thurston geometrization conjectures, mean curvature flow, and minimal submanifolds.
The book is primarily aimed at graduate students in geometric analysis, but it will also be of interest to postdoctoral researchers and established mathematicians looking for a refresher or deeper exploration of the topic.
Reihe
Sprache
Verlagsort
Zielgruppe
Maße
Höhe: 254 mm
Breite: 178 mm
ISBN-13
978-1-4704-7767-7 (9781470477677)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Bennett Chow, University of California, San Diego, La Jolla, CA, and Yutze Chow, University of Wisconsin - Milwaukee, WI.
Geometry of submanifolds of Euclidean space
Intuitive introduction to submanifolds in Euclidean space
Differential calculus of submanifolds
Linearizing submanifolds: Tangent and tensor bundles
Curvature and the local geometry of submanifolds
Global theorems in the theory of submanifolds
Differential topology and Riemannian geometry
Smooth manifolds
Riemannian manifolds
Differential forms and the method of moving frames on manifolds
The Gauss-Bonnet and Poincare-Hopf theorems
Bundles and the Chern-Gauss-Bonnet formula
Elliptic and parabolic equations in geometric analysis
Linear elliptic and parabolic equations
Elliptic equations and the geometry of minimal surfaces
Geometric flows of curves in the plane
Uniformization of surfaces via heat flow
Bibliography
Index
