##### 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

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