Softcover ISBN:  9780821846797 
Product Code:  STML/46 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470418175 
Product Code:  STML/46.E 
List Price:  $49.00 
Individual Price:  $39.20 
Softcover ISBN:  9780821846797 
eBook: ISBN:  9781470418175 
Product Code:  STML/46.B 
List Price:  $108.00 $83.50 
Softcover ISBN:  9780821846797 
Product Code:  STML/46 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470418175 
Product Code:  STML/46.E 
List Price:  $49.00 
Individual Price:  $39.20 
Softcover ISBN:  9780821846797 
eBook ISBN:  9781470418175 
Product Code:  STML/46.B 
List Price:  $108.00 $83.50 

Book DetailsStudent Mathematical LibraryVolume: 46; 2008; 286 ppMSC: Primary 51; 53; 57
Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. At the same time, many of those notions appear in a technically simpler and more graphic form than in their general “natural” settings.
The first, primarily expository, chapter introduces many of the principal actors—the round sphere, flat torus, Möbius strip, Klein bottle, elliptic plane, etc.—as well as various methods of describing surfaces, beginning with the traditional representation by equations in threedimensional space, proceeding to parametric representation, and also introducing the less intuitive, but central for our purposes, representation as factor spaces. It concludes with a preliminary discussion of the metric geometry of surfaces, and the associated isometry groups. Subsequent chapters introduce fundamental mathematical structures—topological, combinatorial (piecewise linear), smooth, Riemannian (metric), and complex—in the specific context of surfaces.
The focal point of the book is the Euler characteristic, which appears in many different guises and ties together concepts from combinatorics, algebraic topology, Morse theory, ordinary differential equations, and Riemannian geometry. The repeated appearance of the Euler characteristic provides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories.
The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background.
This book is a result of the MASS course in geometry in the fall semester of 2007.
This book is published in cooperation with Mathematics Advanced Study Semesters.ReadershipUndergraduate and graduate students interested in broadening their view of geometry and topology.

Table of Contents

Chapters

Chapter 1. Various ways of representing surfaces and basic examples

Chapter 2. Combinatorial structure and topological classification of surfaces

Chapter 3. Differentiable structure on surfaces: Real and complex

Chapter 4. Riemannian metrics and geometry of surfaces

Chapter 5. Topology and smooth structure revisited

Suggested reading

Hints


Additional Material

Reviews

This book will be a welcome addition to college and university libraries and an excellent source for supplementary reading.
Mathematical Reviews 
(This book) does a masterful job of introducing the study of surfaces to advanced undergraduates. ... The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading.
MAA Reviews


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Surfaces are among the most common and easily visualized mathematical objects, and their study brings into focus fundamental ideas, concepts, and methods from geometry, topology, complex analysis, Morse theory, and group theory. At the same time, many of those notions appear in a technically simpler and more graphic form than in their general “natural” settings.
The first, primarily expository, chapter introduces many of the principal actors—the round sphere, flat torus, Möbius strip, Klein bottle, elliptic plane, etc.—as well as various methods of describing surfaces, beginning with the traditional representation by equations in threedimensional space, proceeding to parametric representation, and also introducing the less intuitive, but central for our purposes, representation as factor spaces. It concludes with a preliminary discussion of the metric geometry of surfaces, and the associated isometry groups. Subsequent chapters introduce fundamental mathematical structures—topological, combinatorial (piecewise linear), smooth, Riemannian (metric), and complex—in the specific context of surfaces.
The focal point of the book is the Euler characteristic, which appears in many different guises and ties together concepts from combinatorics, algebraic topology, Morse theory, ordinary differential equations, and Riemannian geometry. The repeated appearance of the Euler characteristic provides both a unifying theme and a powerful illustration of the notion of an invariant in all those theories.
The assumed background is the standard calculus sequence, some linear algebra, and rudiments of ODE and real analysis. All notions are introduced and discussed, and virtually all results proved, based on this background.
This book is a result of the MASS course in geometry in the fall semester of 2007.
Undergraduate and graduate students interested in broadening their view of geometry and topology.

Chapters

Chapter 1. Various ways of representing surfaces and basic examples

Chapter 2. Combinatorial structure and topological classification of surfaces

Chapter 3. Differentiable structure on surfaces: Real and complex

Chapter 4. Riemannian metrics and geometry of surfaces

Chapter 5. Topology and smooth structure revisited

Suggested reading

Hints

This book will be a welcome addition to college and university libraries and an excellent source for supplementary reading.
Mathematical Reviews 
(This book) does a masterful job of introducing the study of surfaces to advanced undergraduates. ... The authors succeed in pulling in many topics while keeping their story coherent and compelling. This book would work well as the text for a capstone course or independent reading.
MAA Reviews