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## Overview

Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, elementary abstract algebra, and point set topology. More than 200 exercises offer students ample opportunity to gauge their skills and gain additional insights.

The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to explore integrating vector fields, Lie bracket, exterior derivative, and Lie derivative. Part III, involving manifolds and vector bundles, develops the main body of the course. The final chapter provides a glimpse into geometric structures by introducing connections on the tangent bundle as a tool to implant the second derivative and the derivative of vector fields on the base manifold. Relevant historical and philosophical asides enhance the mathematical text, and helpful Appendixes offer supplementary material.

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## Product Details

ISBN-13: | 9780486807065 |
---|---|

Publisher: | Dover Publications |

Publication date: | 08/17/2016 |

Series: | Aurora: Dover Modern Math Originals Series |

Edition description: | First Edition, First |

Pages: | 368 |

Sales rank: | 663,842 |

Product dimensions: | 5.90(w) x 9.00(h) x 0.60(d) |

## About the Author

Siavash Shahshahani studied at Berkeley with Steve Smale and received his PhD in 1969, after which he held positions at Northwestern and the University of Wisconsin, Madison. From 1974 until his 2012 retirement he was mainly at Sharif University of Technology in Tehran, Iran, where he helped develop a strong mathematics program.

## Read an Excerpt

#### An Introductory Course on Differentiable Manifolds

**By Siavash Shahshahani**

**Dover Publications, Inc.**

**Copyright © 2016 Siavash Shahshahani**

All rights reserved.

ISBN: 978-0-486-82082-8

All rights reserved.

ISBN: 978-0-486-82082-8

CHAPTER 1

**Multilinear Algebra**

In this chapter, we discuss various structures and mappings that involve one or several vector spaces over a fixed field. It will always be assumed that the field characteristic is zero; in fact, the reader may assume that the underlying field is R or C. In Section D, we will consider special features of vector spaces over R.

**A. Dual Space**

Let *V* be a finite dimensional vector space over a field *F*. The set of linear mappings *V* ->*F* will be denoted by *V**. This set will be endowed with the structure of a vector space over *F*. Let α and β be elements of *V** and *r* an element of *F*, then we define α+β and *r*α by

(1.1) (α+β)(x) = α(x) + β(x)

(1.2) (rα)(x) = rα(x)

where *x* is an arbitrary element of *V*. With these operations, *V** becomes a vector space over *F* and will be called the ** dual space** to

**V**. Suppose (

*e*1, ...,

*en*) is a basis for

*V*. We define elements

*ei*of

*V** by their value on

*ej, j*= 1, ...,

*n*, as follows:

(1.3) ei(ej) = σij

where σ*i* denotes the value 1 or 0 depending on whether *i=j* or *i* ≠ *j*. Note that any element α of *V** can be written as a linear combination of *e*1, ..., *en*. In fact,

[MATHEMATICAL EXPRESSION OMITTED]

since the value of both sides on an arbitrary basis element *ej* is the same. Further, {*e*1, ..., *en*} is a linearly independent set, for if [summation]*iriei*=0, evaluating both sides on the basis element *ej* yields *rj*=0. Therefore, the ordered set (*e*1, ..., *en*) is a basis for *V**, called the dual basis for *V** relative to (*e*1, ..., *en*). Thus *V** has the same dimension as *V*.

By repeating the operation of dual making, one can look at (*V**)*, the so-called ** double dual** of

*V*, usually denoted by

*V***. The double dual will then have the same dimension as the original space

*V*, and since all linear spaces of the same dimension over a given field are isomorphic, there are isomorphisms between

*V, V** and

*V***. But in the case of

*V*and

*V***, there is a distinguished

**, denoted by**

*natural isomorphism**IV:V*->

*V***, which is given as follows. For each

*v*[member of]

*V*, the element

*IV(v)*is defined by

(1.4) (*IV*(v))(α) = α(v)

It follows from (1.1) and (1.2) that *IV(v)* is indeed linear, i.e., it is a member of *V***. That *IV* is linear follows from the linearity of α. To show that *IV* is an isomorphism, it suffices to show that its kernel is {0} since the domain and target linear spaces are finite dimensional of the same dimension. But α(*v*)=0 for all α in *V** implies that *v*=0, and the isomorphism is established. Note that the definition of *IV* was independent of the specific nature of the linear space *V* or the choice of basis for it. In fact, one can state the following general assertion.

**1. Theorem***For any basis* (*e*1, ..., *en*) *of V*, (*IV* (*e*1), ..., *IV(en])ITL) is the dual basis in V** relative to the basis (e1, ..., en) for V*.*

*Proof. We must show*

*[MATHEMATICAL EXPRESSION OMITTED]*

*This is a consequence of (1.4) and (1.3).*

By virtue of the natural isomorphism *IV*, the space *V*** is often identified with *V*. Under this identification, *IV*(*ei*) is identified with *ei*, so that (*e*1, ..., *en*) becomes the dual basis for *V*** relative to (*e*1, ..., *en*).

**B. Tensors**

Let *V*1, ..., *Vp* and *W* be vector spaces over a field. A map α: *V*1 X ... X *Vp* ->*W* is called ** p-linear** provided that by fixing any

*p*-1 components of (

*v*1, ...,

*vp*) [member of]

*V*1 X ... X

*Vp*, α is linear with respect to the remaining component. As we shall see in some of the following examples, operations generally known as "products" in elementary mathematics are of this nature.

2. Examples

(a) Let *V* be a vector space over a field *F*. Regard *F* as a one-dimensional vector space over *F*. Then the product *F* X *V* ->*V* given by

(r; v) [??] rv

is 2-linear (** bilinear**).

(b) Let *F* be a field. Then the *p*-fold product *F* X ... X *F* ->*F* given by

(r1, ..., rp) [??] r1 ... rp

is *p*-linear.

(c) Let *V* be a vector space over R. Then any inner product *V* X *V* -> R is bilinear. The vector product R3 X R3 -> R3 is another example of a bilinear mapping. In general, let β: *V* X *V* ->*F* be bilinear and consider a basis (*e*1, ..., *en*) for *V*. The *n*X*n* matrix B=[β*ij*], where β*ij* = β(*ei; ej*), determines β completely as

(1.5) [MATHEMATICAL EXPRESSION OMITTED]

If B is a symmetric matrix with positive eigenvalues, then β is an inner product.

Conversely, any inner product on V is obtained in this manner.

(d) For a vector space *V* over a field *F*, the * evaluation pairing V**

*V** ->

*F*, given by (

*v*, α) [??]α (

*v*), is bilinear.

(e) Let *F* be a field and *V*1, ..., *Vp; W*1, ..., *Wq* be vector spaces over *F*. Suppose p-linear and *q*-linear maps α:*V*1 X ... *Vp* ->*F* and β:*W*1 X ... X *Wq* -> F are given. Then the tensor product

[MATHEMATICAL EXPRESSION OMITTED]

is defined by

(1.6) [MATHEMATICAL EXPRESSION OMITTED]

Note that α [cross product] β is a (*p+q*)-linear mapping. Further, it follows from the associativity of the product operation in the field *F* that is associative, hence the product α1 [cross product] ... [cross product] α*k* is unambiguously defined by induction.

In what follows, *V* will be a finite dimensional vector space over a field *F*. The *n*-fold product *V* X ... X *V* will be denoted by *Vn*.

**3. Definition**

(a) A *p*-linear map *Vp* ->*F* will be called a ** covariant p-tensor**, or a

**, on**

*tensor of type (p,0)**V*.

(b) A *q*-linear map (*V**)*q* ->*F* will be called a ** contravariant q-tensor**, or a

**, on**

*tensor of type (0,q)**V*.

(c) A (*p + q*)-linear map *Vp*(*V**)q ->*F* will be called a ** mixed (p,q)-tensor**, or a

**, on**

*tensor of type (p,q)**V*.

**4. Examples** An element of *V** is a covariant 1-tensor on *V*. In view of the natural isomorphism *IV*, any member of *V* may be regarded as a contravariant tensor on *V*. The evaluation pairing (Example 2d) is a (1; 1)-tensor on *V*. Inner products are examples of covariant 2-tensors.

We use the symbols *Lp(V), Lq(V)* and *Lpq (V)*, respectively, to denote the sets of (*p*; 0)-, (0; *q*)- and (*p; q*)-tensors on *V*. Under functional addition, and multiplication by elements of the field *F*, each of these becomes a vector space over *F*. The dimensions of these spaces are, respectively, *np, nq* and *np+q*, as the following will imply.

**5. Basis for the Space of Tensors***Let* (*e*1, ..., *en*) *be a basis for V. Then the following are basis elements for the spaces of tensors*.

(a) *For Lp(V):*

(1.7) [MATHEMATICAL EXPRESSION OMITTED]

(b) *For Lq(V):*

(1.8) [MATHEMATICAL EXPRESSION OMITTED]

(c) *For Lpq (V):*

(1.9) [MATHEMATICAL EXPRESSION OMITTED]

Proof. Note that by virtue of Example 2e, the displayed tensors are actually elements of the stated spaces. We prove the third case which includes the other two. To show linear independence, suppose that

[MATHEMATICAL EXPRESSION OMITTED]

By applying the two sides to ([MATHEMATICAL EXPRESSION OMITTED]), we see that the coefficients are zero, and linear independence is established. On the other hand, note that any α [member of] *Lpq* can be written as

(1.10) [MATHEMATICAL EXPRESSION OMITTED]

which can be verified by applying both sides to ([MATHEMATICAL EXPRESSION OMITTED]).

By convention, we let *L*0*V=L*0*V=F*.

**6. Change of Basis**

The bases introduced above for the spaces of tensors as well as the resulting components of the tensors depend on the original choice of basis for the linear space.

We are now going to investigate how a linear change of basis for the space affects the value of tensor components. We take *V* to be an *n*-dimensional vector space over *F*. It will be convenient to write *n*X*n* matrices with entries from *F* as A=[*aij*], where the superscript denotes the row index and the subscript indicates the column of the matrix entry. Suppose two bases B=(*e*1, ..., *en*) and [bar.B] = ([bar.*e*]1, ..., *[bar.e]n*) are given for *V*, related linearly by matrix A=[*aij*] as

(1.11) [MATHEMATICAL EXPRESSION OMITTED]

Thus the components of *[bar.e]j* with respect to the basis B are the entries of the *j*th column of matrix A. Corresponding to B and [bar.B], we have the dual bases B*=(*e*1, ..., *en*) and [bar.B]*=([bar.*e*]1, ..., *[bar.e]n*). We will first investigate the linear relationship between these two bases. We write

(1.12) [MATHEMATICAL EXPRESSION OMITTED]

Therefore, the components of *[bar.e]i* with respect to the basis B* are the entries of the ith row of matrix B=[*bij*]. To identify B, we note that

[MATHEMATICAL EXPRESSION OMITTED]

Therefore, the matrix B is the inverse of the transpose of the matrix A:

B-1 = AT

Now let α be a (*p, q*)-tensor on *V*. With respect to the above bases, the following two representations for α are obtained.

[MATHEMATICAL EXPRESSION OMITTED]

We wish to express the components [MATHEMATICAL EXPRESSION OMITTED] in terms of [MATHEMATICAL EXPRESSION OMITTED]. Using (1.10), we have

[MATHEMATICAL EXPRESSION OMITTED]

This is equal to

[MATHEMATICAL EXPRESSION OMITTED]

Thus we have obtained the desired formula for the change of tensor components under a linear change of variables

(1.13) [MATHEMATICAL EXPRESSION OMITTED]

Classically, a tensor is defined as a collection of numerical quantities [MATHEMATICAL EXPRESSION OMITTED] which transform under a linear change of variables according to formula (1.13).

(a) **Special case (p=1,q=0)** For a covariant 1-tensor

[MATHEMATICAL EXPRESSION OMITTED]

we obtain

(1.14) [MATHEMATICAL EXPRESSION OMITTED]

(b) **Special case (p=0,q=1)** Consider a contravariant 1-tensor, or by virtue of the natural isomorphism *IV*, an element *x* of *V*

[MATHEMATICAL EXPRESSION OMITTED]

In this case, we have

(1.15) [MATHEMATICAL EXPRESSION OMITTED]

**7. Functoriality**

Let *V* and *W* be vector spaces over a field *F*, and suppose *f:V* ->*W* is a linear map. For each non-negative integer *p*, a map *Lpf:LpW* ->*LpV* is defined as follows. If *p* = 0, *L*0*f*=1*F*. For p > 0, suppose α [member of] *LpW* and *v*1, ..., *vp* [member of] *V*, then

(1.16) [MATHEMATICAL EXPRESSION OMITTED]

That (*Lp f*)(α) [member of] *LpV* follows from the linearity of *f* and the fact that α [member of] *LpW*. The linearity of Lp f follows from the definition of linear space operations in the space of tensors. The following two properties are straightforward consequences of definition and establish Lp as a contravariant functor.

(a) *For any vector space V and any non-negative integer p,*

(1.17) [MATHEMATICAL EXPRESSION OMITTED]

(b) *For linear maps f : V* ->*W and g : U* ->*V, and any non-negative integer p*,

(1.18) Lp(f ο g) = Lpg ο Lp f

Of course, *L*1*V* = *V**. The induced linear map *L*1*f* is denoted by *f**. Note that by definition, *LqV** = *LqV*. For a linear map *f : V* ->*W*, we denote *Lqf** by *Lqf*. The following properties follow from (a) and (b) above and are summarized by saying that *Lq* is a ** covariant functor**.

(c) *For any vector space V and non-negative integer q,*

(1.19) [MATHEMATICAL EXPRESSION OMITTED]

(d) *For linear maps f : V* ->*W and g : U* ->*V, and any non-negative integer q,*

(1.20) Lq(f ο g) = Lq f ο Lqg

**C. Anti-symmetric Tensors**

The so-called *anti-symmetric tensors* are among the most powerful tools in the study of geometric structures. As we shall see in the following section, these are closely related to the concepts of volume and orientation in the case of real vector spaces.

We recall some elementary facts about the group S*n* of permutations on *n* symbols {1, ..., *n*}. A ** transposition** is a permutation that exchanges two symbols and leaves the other symbols fixed. Any permutation σ [member of] S

*n*can be written as a composition of transpositions, σ_=_τ1 ο ... οτ

*k*, where

*k*is not unique but its parity (even- or oddness) is determined by σ. Thus a permutation σ is called

**or**

*even***depending on whether**

*odd**k*is even or odd. We write ε(σ) = +1 or ε(σ) = -1, respectively, if σ is even or odd. The map ε, called the

**, is a homomorphism from S**

*sign**n*onto the two-element multiplicative group {+1; -1}; thus ε(σ1 ο σ2)=ε(σ1) ο ε(σ2) and [MATHEMATICAL EXPRESSION OMITTED]. The set of even permutations form a subgroup of index 2 in S

*n*.

*(Continues...)*

Excerpted fromAn Introductory Course on Differentiable ManifoldsbySiavash Shahshahani. Copyright © 2016 Siavash Shahshahani. Excerpted by permission of Dover Publications, Inc..

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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*
*

## Table of Contents

Preface 1

Part I Pointwise 5

Chapter 1 Multilinear Algebra 7

A Dual Space 7

B Tensors 8

C Anti-symmetric Tensors 13

D Real Linear Spaces 17

E Product Structure 19

Exercises 24

Part II Local 27

Chapter 2 Vector Fields: Local Theory 29

A Tangent Space 29

B Vector Fields and Differential Equations 33

C Vector Fields as Operators 44

D Lie Bracket of Vector Fields 51

Exercises 60

Chapter 3 Tensor Fields: Local Theory 65

A Basic Constructions 65

B Pointwise Operations 68

C Exterior Derivative 72

D Lie Derivative 77

E Riemannian Metrics 81

Exercises 86

Part III Global 91

Chapter 4 Manifolds, Tangent Bundle 93

A Topological Manifolds 93

B Smooth Manifolds 102

C Smooth Structures 104

D The Tangent Bundle 109

Appendix 120

Exercises 122

Chapter 5 Mappings, Submanifolds and Quotients 125

A Submanifolds 125

B Immersions, Submersions and Embeddings 132

C Quotient Manifolds 137

D Covering Spaces 142

Exercises 150

Chapter 6 Vector Bundles and Fields 155

A Basic Constructions 155

B Vector Fields: Globalization 164

C Differential Forms: Globalization 169

D Riemannian Metrics 175

E Plane Fields 181

Exercises 195

Chapter 7 Integration and Cohomology 203

A Manifolds with Boundary 203

B Integration on Manifolds with Boundary 209

C Stokes Theorem 217

D De Rham Cohomology 222

E Top-dimensional Cohomology and Applications 232

Exercises 242

Chapter 8 Lie Groups and Homogeneous Spaces 251

A Continuous Groups 251

B The Lie Algebra of a Lie Group 261

C Homogeneous Spaces 271

Exercises 279

Part IV Geometric Structures 285

Chapter 9 An Introduction to Connections 287

A The Geography of the Double Tangent Bundle 287

B Descent of the Second Derivative 295

C Covariant Derivative 303

D Cutvature and Torsion 310

E Newtonian Mechanics 321

Exercises 330

Appendix I The Exponential of a Matrix 337

Appendix II Differential Calculus in Normed Space 341

Bibliography 349

List of symbols 351

Index 353

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