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TitleA First Course in Geometric Topology and Differential Geometry
File Size8.9 MB
Total Pages440
Table of Contents
                            Cover
Title Page
Contents
Introduction
To the Student
	Surfaces
	Prerequisites
	Rigor vs. Intuition
	Exercises
I. Topology of Subsets of Euclidean Space
	1.1. Introduction
	1.2. Open and Closed Subsets of Sets in ℝⁿ
	1.3. Continuous Maps
	1.4. Homeomorphisms and Quotient Maps
	1.5. Connectedness
	1.6. Compactness
	Endnotes
II. Topological Surfaces
	2.1 Introduction
	2.2 Arcs, Disks and 1-Spheres
	2.3 Surfaces in ℝⁿ
	2.4 Surfaces via Gluing
	2.5 Properties of Surfaces
	2.6 Connected Sum and the Classification of Compact Connected Surfaces
	Appendix A2.1 Proof of Theorem 2.4.3 (i)
	Appendix A2.2 Proof of Proposition 2.6.1
	Endnotes
III. Simplicial Surfaces
	3.1 Introduction
	3.2 Simplices
	3.3 Simplicial Complexes
	3.4 Simplicial Surfaces
	3.5 The Euler Characteristic
	3.6 Proof of the Classification of Compact Connected Surfaces
	3.7 Simplicial Curvature and the Simplicial Gauss-Bonnet Theorem
	3.8 Simplicial Disks and the Brouwer Fixed Point Theorem
	Endnotes
IV. Curves in ℝ³
	4.1 Introduction
	4.2. Smooth Functions
	4.3 Curves in ℝ³
	4.4 Tangent, Normal and Binormal Vectors
	4.5 Curvature and Torsion
	4.6 Fundamental Theorem of Curves
	4.7 Planar Curves
	Endnotes
V. Smooth Surfaces
	5.1 Introduction
	5.2 Coordinate Patches and Smooth Surfaces
	5.3 Examples of Smooth Surfaces
	5.4 Tangent and Normal Vectors
	5.5 First Fundamental Form
	5.6 Directional Derivatives — Coordinate-Free
	5.7 Directional Derivatives — Coordinates
	5.8 Length and Area
	5.9 Isometries
	Appendix A5.1 Proof of Proposition 5.3.1
	Endnotes
VI. Curvature of Smooth Surfaces
	6.1 Introduction
	6.2 The Weingarten Map and the Second Fundamental Form
	6.3 Curvature — Second Attempt
	6.4 Computations of Curvature Using Coordinates
	6.5 Theorema Egregium and the Fundamental Theorem of Surfaces
	Endnotes
VII. Geodesics
	7.1 Introduction
	7.2 Geodesics
	7.3 Shortest Paths
	Endnotes
VIII. The Gauss-Bonnet Theorem
	8.1 Introduction
	8.2 The Exponential Map
	8.3 Geodesic Polar Coordinates
	8.4 Proof of the Gauss-Bonnet Theorem
	8.5 Non-Euclidean Geometry
	Appendix A8.1 Geodesic Convexity
	Appendix A8.2 Geodesic Triangulations
	Endnotes
Appendix
	Affine Linear Algebra
Further Study
	1. Collateral Reading
	2. Point Set Topology (also known as General Topology)
	3. Algebraic Topology
	4. Geometric Topology
	5. Differential Geometry
	6. Differential Topology
References
Hints for Selected Exercises
	Section 1.2
	Section 1.3
	Section 1.4
	Section 1.5
	Section 1.6
	Section 2.2
	Section 2.3
	Section 2.4
	Section 2.5
	Section 2.6
	Appendix A2.2
	Section 3.2
	Section 3.3
	Section 3.4
	Section 3.5
	Section 3.7
	Section 3.8
	Section 4.2
	Section 4.3
	Section 4.4
	Section 4.5
	Section 4.6
	Section 4.7
	Section 5.2
	Section 5.3
	Section 5.4
	Section 5.5
	Section 5.6
	Section 5.7
	Section 5.9
	Section 6.1
	Section 6.2
	Section 6.3
	Section 6.4
	Section 6.5
	Section 7.2
	Section 8.2
	Section 8.3
	Appendix A8.1
	Appendix A8.2
Index of Notation
Index
Back Cover
                        
Document Text Contents
Page 1

0 A FirstCourse in
Geometric
Topology
and
Differential
Geometry

Ethan D. Bloch

i
Birkhauser

Page 220

204 V. Smooth Surfaces

To see that y is injective, observe that y((; )) = y((" )) implies s3 = u3 and
t 3 = v3, and hence s = u and t = v. The partial derivatives of y are

3s2

)' = 0
0

and

0

y2 =
(32).

0

Hence

0

y1xy2= 0
9s2t2

which is zero whenever s = 0 or t = 0. Therefore y is not a coordinate patch.
0

Coordinate patches now allow us to define smooth surfaces.

Definition. A subset M C R3 is a smooth surface if it is a topological surface
and if for each point p E M there is a coordinate patch x: U -+ M C R3 such
thatpEx(U). 0

In practice, rather than finding a coordinate patch for each point p in a
smooth surface we simply find coordinate patches whose images cover the
entire surface. In many cases more than one coordinate patch will be needed.
We will not give explicit proofs that the surfaces under consideration are indeed
topological surfaces, since it will usually be quite straightforward.

Example 5.2.2. (1) Any open subset U C R2 is a smooth surface covered by
the coordinate patch x: U -+ R3 given by

s

t

(2) The unit sphere S2 is a smooth surface. One method to cover S2 with coordi-
nate patches is to use the six coordinate patches x1,x2, yt, y2, zi, z2: int D2-+

Page 221

5.2 Coordinate Patches and Smooth Surfaces

S2 given by

205

Each of these coordinate patches covers an open hemisphere; see Figure 5.2.1.
We leave it to the reader to verify that these six maps are actually coordinate
patches. 0

Figure 5.2.1

ny I

What is the relation between smooth surfaces, topological surfaces and
simplicial surfaces? By definition any smooth surface is a topological surface.
It then follows from Theorem 3.4.5 that every compact smooth surface can be
triangulated. Is every topological surface also a smooth surface? Whereas not

Page 440

Ethan D. Bloch

A First Course in
Geometric Topology
and Differential Geometry
The uniqueness of this text in combining geometric
topology and differential geometry lies in its unifying

thread: the notion of a surface. With numerous illus-
trations, exercises and examples, the student comes

to understand the relationship of the modern abstract
approach to geometric intuition. The text is kept at a

concrete level, avoiding unnecessary abstractions,
yet never sacrificing mathematical rigor. The book
includes topics not usually found in a single book at

this level.

A number of intuitively appealing definitions and
theorems concerning surfaces in the topological,
polyhedral and smooth cases are presented from
the geometric view. Point set topology is restricted to
subsets of Euclidean spaces. The treatment of differ-

ential geometry is classical, dealing with surfaces in

R3. Included are the classification of compact sur-
faces, the Gauss-Bonnet Theorem and the geodesic

nature of length minimizing curves on surfaces.

The material here should be accessible to math
majors at the junior/senior level in an American
university or college, the minimal prerequisites
being standard Calculus sequence (including multi-
variable Calculus and an acquaintance with differen-

tial equations), linear algebra (including inner prod-
ucts), and familiarity with proofs and the basics of sets

and functions.

ISBN 0-8376-3840-7

ISBN 0-8176-3840-7

0

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