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An Introduction to Differential Geometry Through Computation

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An Introduction to Differential Geometry through Computation Mark E. Fels c Draft date April 18, 2011 Contents Preface iii 1 Preliminaries 1 1.1 Open sets . .1 1.2 Smooth functions . .2 1.3 Smooth Curves . .5 1.4 Composition and the Chain-rule . .6 1.5 Vector Spaces, Basis, and Subspaces . .9 1.6 Algebras . 17 1.7 Exercises . 20 2 Linear Transformations 25 2.1 Matrix Representation . 25 2.2 Kernel, Rank, and the Rank-Nullity Theorem . 30 2.3 Composition, Inverse and Isomorphism . 34 2.4 Exercises . 39 3 Tangent Vectors 43 3.1 Tangent Vectors and Curves . 43 3.2 Derivations . 44 3.3 Vector fields . 51 3.4 Exercises . 53 4 The push-forward and the Jacobian 55 4.1 The push-forward using curves . 55 4.2 The push-forward using derivations . 58 4.3 The Chain rule, Immersions, Submersions, and Diffeomorphisms 62 4.4 Change of Variables . 65 i ii CONTENTS 4.5 Exercises . 71 5 Differential One-forms and Metric Tensors 75 5.1 Differential One-Forms . 75 5.2 Bilinear forms and Inner Products . 81 5.3 Tensor product . 86 5.4 Metric Tensors . 89 5.4.1 Arc-length . 93 5.4.2 Orthonormal Frames . 95 5.5 Raising and Lowering Indices and the Gradient . 97 5.6 A tale of two duals . 105 5.7 Exercises . 107 6 The Pullback and Isometries 111 6.1 The Pullback of a Differential One-form . 111 6.2 The Pullback of a Metric Tensor . 117 6.3 Isometries . 122 6.4 Exercises . 130 10 Hypersurfaces 187 10.1 Regular Level Hyper-Surfaces . 187 10.2 Patches and Covers . 189 10.3 Maps between surfaces . 192 10.4 More General Surfaces . 193 10.5 Metric Tensors on Surfaces . 193 10.6 Exercises . 200 8 Flows, Invariants and the Straightening Lemma 149 8.1 Flows . 149 8.2 Invariants . 155 8.3 Invariants I . 158 8.4 Invariants II . 163 8.5 Exercises . 168 9 The Lie Bracket and Killing Vectors 171 9.1 Lie Bracket . 171 9.2 Killing vectors . 179 9.3 Exercises . 185 CONTENTS iii 10 Hypersurfaces 187 10.1 Regular Level Hyper-Surfaces . 187 10.2 Patches and Covers . 189 10.3 Maps between surfaces . 192 10.4 More General Surfaces . 193 10.5 Metric Tensors on Surfaces . 193 10.6 Exercises . 200 11 Group actions and Multi-parameter Groups 203 11.1 Group Actions . 203 11.2 Infinitesimal Generators . 207 11.3 Right and Left Invariant Vector Fields . 209 11.4 Invariant Metric Tensors on Multi-parameter groups . 211 11.5 Exercises . 214 12 Connections and Curvature 217 12.1 Connections . 217 12.2 Parallel Transport . 217 12.3 Curvature . 217 iv CONTENTS Preface This book was conceived after numerous discussions with my colleague Ian Anderson about what to teach in an introductory one semester course in differential geometry. We found that after covering the classical differential geometry of curves and surfaces that it was difficult to make the transition to more advanced texts in differential geometry such as [?], or to texts which use differential geometry such as in differential equations [?] or general relativity [?], [?]. This book aims to make this transition more rapid, and to prepare upper level undergraduates and beginning level graduate students to be able to do some basic computational research on such topics as the isometries of metrics in general relativity or the symmetries of differential equations. This is not a book on classical differential geometry or tensor analysis, but rather a modern treatment of vector fields, push-forward by mappings, one-forms, metric tensor fields, isometries, and the infinitesimal generators of group actions, and some Lie group theory using only open sets in IR n. The definitions, notation and approach are taken from the corresponding concept on manifolds and developed in IR n. For example, tangent vectors are defined as derivations (on functions in IR n) and metric tensors are a field of positive definite symmetric bilinear functions on the tangent vectors. This approach introduces the student to these concepts in a familiar setting so that in the more abstract setting of manifolds the role of the manifold can be emphasized. The book emphasizes liner algebra. The approach that I have taken is to provide a detailed review of a linear algebra concept and then translate the concept over to the field theoretic version in differential geometry. The level of preparation in linear algebra effects how many chapters can be covered in one semester. For example, there is quite a bit of detail on linear transformations and dual spaces which can be quickly reviewed for students with advanced training in linear algebra. v CONTENTS 1 The outline of the book is as follows. Chapter 1 reviews some basic facts about smooth functions from IR n to IR m, as well as the basic facts about vector spaces, basis, and algebras. Chapter 2 introduces tangent vec- tors and vector fields in IR n using the standard two approaches with curves and derivations. Chapter 3 reviews linear transformations and their matrix representation so that in Chapter 4 the push-forward as an abstract linear transformation can be defined and its matrix representation as the Jacobian can be derived. As an application, the change of variable formula for vector fields is derived in Chapter 4. Chapter 5 develops the linear algebra of the dual space and the space of bi-linear functions and demonstrates how these concepts are used in defining differential one-forms and metric tensor fields. Chapter 6 introduces the pullback map on one-forms and metric tensors from which the important concept of isometries is then defined. Chapter 7 inves- tigates hyper-surfaces in IR n, using patches and defines the induced metric tensor from Euclidean space. The change of coordinate formula on overlaps is then derived. Chapter 8 returns to IR n to define a flow and investigates the relationship between a flow and its infinitesimal generator. The theory of flow invariants is then investigated both infinitesimally and from the flow point of view with the goal of proving the rectification theorem for vector fields. Chapter 9 investigates the Lie bracket of vector-fields and Killing vec- tors for a metric. Chapter 10 generalizes chapter 8 and introduces the general notion of a group action with the goal of providing examples of metric tensors with a large number of Killing vectors. It also introduces a special family of Lie groups which I've called multi-parameter groups. These are Lie groups whose domain is an open set in IR n. The infinitesimal generators for these groups are used to construct the left and right invariant vector-fields on the group, as well as the Killing vectors for some special invariant metric tensors on the groups. 2 CONTENTS Chapter 1 Preliminaries 1.1 Open sets The components (or Cartesian coordinates ) of a point x 2 IR n will be denoted by x = (x1; x2; : : : ; xn): Note that the labels are in the up position. That is x2 is not the square of x unless we are working in IR 1; IR 2; IR 3 where we will use the standard notation of x; y; z. The position of indices is important, and make many formulas easier to remember or derive. The Euclidean distance between the points x = (x1; : : : ; xn) and y = (y1; : : : ; yn) is d(x; y) = p(x1 − y1)2 + ::: + (xn − yn)2: + n The open ball of radius r 2 IR at the point p 2 IR is the set Br(p) ⊂ IR n, defined by n Br(p) = f x 2 IR j d(x; p) < rg: A subset U ⊂ IR n is an open set if given any point p 2 U there exists + an r 2 IR (which depends on p) such that the open ball Br(p) satisfies Br(p) ⊂ U. The empty set is also taken to be open. Example 1.1.1. The set IR n is an open set. n + Example 1.1.2. Let p 2 IR and r 2 IR . Any open ball Br(p) is an open set. 1 2 CHAPTER 1. PRELIMINARIES Example 1.1.3. The upper half plane is the set U = f (x; y) 2 IR 2 j y > 0 g and is open. Example 1.1.4. The set V = f (x; y) 2 IR 2 j y ≥ 0 g is not open. Any point (x; 0) 2 V can not satisfy the open ball condition. Example 1.1.5. The unit n-sphere Sn ⊂ IR n+1 is the subset Sn = f x 2 IR n+1 j d(x; 0) = 1 g and Sn is not open. No point x 2 Sn satisfies the open ball condition. The n n+1 set S is the boundary of the open ball B1(0) ⊂ IR . Roughly speaking, open sets contain no boundary point. This can be made precise using some elementary topology. 1.2 Smooth functions In this section we recall some facts from multi-variable calculus. A real-valued function f : IR n ! IR has the form, f(x) = f(x1; x2; : : : ; xn): We will only be interested in functions whose domain Dom(f), is either all of IR n or an open subset U ⊂ IR n. For example f(x; y) = log xy is defined only on the set U = f(x; y) 2 IR 2 j xy > 0g, which is an open set in IR 2, and Dom(f) = U. A function f : IR n ! IR is continuous at p 2 IR n if lim f(x) = f(p): x!p If U ⊂ IR n is an open set, then C0(U) denotes the functions defined on U which are continuous at every point of U.
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