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An Introductory Course on Differentiable Manifolds
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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 fourpart 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.
Product Details
ISBN13:  9780486807065 

Publisher:  Dover Publications 
Publication date:  08/17/2016 
Series:  Aurora: Dover Modern Math Originals 
Edition description:  First Edition, First 
Pages:  368 
Sales rank:  629,054 
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 ShahshahaniAll rights reserved.
ISBN: 9780486820828
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 rα 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 (e1, ..., 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 σi 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 e1, ..., en. In fact,
[MATHEMATICAL EXPRESSION OMITTED]
since the value of both sides on an arbitrary basis element ej is the same. Further, {e1, ..., 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 (e1, ..., en) is a basis for V*, called the dual basis for V* relative to (e1, ..., en). Thus V* has the same dimension as V.
By repeating the operation of dual making, one can look at (V*)*, the socalled 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 natural isomorphism, denoted by 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. TheoremFor any basis (e1, ..., en) of V, (IV (e1), ..., 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 (e1, ..., en) becomes the dual basis for V** relative to (e1, ..., en).
B. Tensors
Let V1, ..., Vp and W be vector spaces over a field. A map α: V1 X ... X Vp >W is called plinear provided that by fixing any p1 components of (v1, ..., vp) [member of] V1 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 onedimensional vector space over F. Then the product F X V >V given by
(r; v) [??] rv
is 2linear (bilinear).
(b) Let F be a field. Then the pfold product F X ... X F >F given by
(r1, ..., rp) [??] r1 ... rp
is plinear.
(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 (e1, ..., en) for V. The nXn matrix B=[βij], where βij = 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 V1, ..., Vp; W1, ..., Wq be vector spaces over F. Suppose plinear and qlinear maps α:V1 X ... Vp >F and β:W1 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] αk is unambiguously defined by induction.
In what follows, V will be a finite dimensional vector space over a field F. The nfold product V X ... X V will be denoted by Vn.
3. Definition
(a) A plinear map Vp >F will be called a covariant ptensor, or a tensor of type (p,0), on V.
(b) A qlinear map (V*)q >F will be called a contravariant qtensor, or a tensor of type (0,q), on V.
(c) A (p + q)linear map Vp(V*)q >F will be called a mixed (p,q)tensor, or a tensor of type (p,q), on V.
4. Examples An element of V* is a covariant 1tensor 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 2tensors.
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 TensorsLet (e1, ..., 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 L0V=L0V=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 ndimensional vector space over F. It will be convenient to write nXn 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=(e1, ..., 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 jth column of matrix A. Corresponding to B and [bar.B], we have the dual bases B*=(e1, ..., 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:
B1 = 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 1tensor
[MATHEMATICAL EXPRESSION OMITTED]
we obtain
(1.14) [MATHEMATICAL EXPRESSION OMITTED]
(b) Special case (p=0,q=1) Consider a contravariant 1tensor, 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 nonnegative integer p, a map Lpf:LpW >LpV is defined as follows. If p = 0, L0f=1F. For p > 0, suppose α [member of] LpW and v1, ..., 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 nonnegative integer p,
(1.17) [MATHEMATICAL EXPRESSION OMITTED]
(b) For linear maps f : V >W and g : U >V, and any nonnegative integer p,
(1.18) Lp(f ο g) = Lpg ο Lp f
Of course, L1V = V*. The induced linear map L1f 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 nonnegative integer q,
(1.19) [MATHEMATICAL EXPRESSION OMITTED]
(d) For linear maps f : V >W and g : U >V, and any nonnegative integer q,
(1.20) Lq(f ο g) = Lq f ο Lqg
C. Antisymmetric Tensors
The socalled antisymmetric 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 Sn 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] Sn can be written as a composition of transpositions, σ_=_τ1 ο ... οτk, where k is not unique but its parity (even or oddness) is determined by σ. Thus a permutation σ is called even or odd depending on whether 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 sign, is a homomorphism from Sn onto the twoelement 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 Sn.
(Continues...)
Excerpted from An Introductory Course on Differentiable Manifolds by Siavash Shahshahani. Copyright © 2016 Siavash Shahshahani. Excerpted by permission of Dover Publications, Inc..
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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 Antisymmetric 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 Topdimensional 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