Hardcover ISBN:  9781470477677 
Product Code:  GSM/245 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Softcover ISBN:  9781470478049 
Product Code:  GSM/245.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470478032 
Product Code:  GSM/245.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470478049 
eBook: ISBN:  9781470478032 
Product Code:  GSM/245.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 
Hardcover ISBN:  9781470477677 
Product Code:  GSM/245 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Softcover ISBN:  9781470478049 
Product Code:  GSM/245.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470478032 
Product Code:  GSM/245.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470478049 
eBook ISBN:  9781470478032 
Product Code:  GSM/245.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsGraduate Studies in MathematicsVolume: 245; 2024; 750 ppMSC: Primary 53; 57; 58
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 Poincaré 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.
ReadershipUndergraduate and graduate students and researchers interested in differential geometry and geometric analysis.

Table of Contents

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 Poincaré–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


Additional Material

RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
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 Poincaré 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.
Undergraduate and graduate students and researchers interested in differential geometry and geometric analysis.

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 Poincaré–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