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Elementary Differential Geometry

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Elementary Differential Geometry Lecture Notes for Differential Geometry James S. Cook Liberty University Department of Mathematics Summer 2015 2 Contents 1 introduction 5 1.1 points and vectors . .5 1.2 on tangent and cotangent spaces and bundles . .7 1.3 the wedge product and differential forms . 11 1.3.1 the exterior algebra in three dimensions . 15 1.4 paths and curves . 16 1.5 the push-forward or differential of a map . 19 2 curves and frames 25 2.1 on distance in three dimensions . 26 2.2 vectors and frames in three dimensions . 27 2.3 calculus of vectors fields along curves . 33 2.4 Frenet Serret frame of a curve . 35 2.4.1 the non unit-speed case . 41 2.5 covariant derivatives . 44 2.6 frames and connection forms . 48 2.6.1 on matrices of differential forms . 49 2.7 coframes and the Structure Equations of Cartan . 53 3 euclidean geometry 57 3.1 isometries of euclidean space . 58 3.2 how isometries act on vectors . 61 n 3.2.1 Frenet curves in R ................................ 63 3.3 on frames and congruence in three dimensions . 67 introduction and motivations for these notes Certainly many excellent texts on differential geometry are available these days. These notes most closely echo Barrett O'neill's classic Elementary Differential Geometry revised second edition. I taught this course once before from O'neil's text and we found it was very easy to follow, however, I will diverge from his presentation in several notable ways this summer. 1. I intend to use modern notation for vector fields. I will resist the use of bold characters as I find it frustrating when doing board work or hand-written homework. 2. I will make some general n-dimensional comments here and there. So, there will be two tracks 3 in these notes: first, the direct extension of typical American third semester calculus in R 3 4 CONTENTS (with the scary manifold-theoretic notation) and second, some results and thoughts for the n-dimensional context. I hope to borrow some of the wisdom of Wolfgang K¨uhnel's Differential Geometry: Curves-Surfaces- Manifolds. I think the purely three dimensional results are readily acessible to anyone who has taken third semester calculus. On the other hand, general n-dimensional results probably make more sense if you've had a good exposure to abstract linear algebra. I do not expect the student has seen advanced calculus before studying these notes. However, on the other hand, I will defer proofs of certain claims to our course in advanced calculus. Chapter 1 introduction n These notes largely concern the geometry of curves and surfaces in R . I try to use a relatively modern notation which should allow the interested student a smooth1 transition to further study of abstract manifold theory. That said, most of what I do in this chapter is merely to dress multi- variate analysis in a new notation. In particular, I decided to sacrifice the pedagogy of O'neill's text in part here; I try to intro- duce notations which are not transitional in nature. For example, I introduce coordinates with contravariant indices and we adopt the derivation formulation of tangent vectors early in our discussion. The tangent bundle and space are openly discussed. However, we do not digress too far into bundles. The purpose of this chapter is primarily notational, if you had advanced calculus then it would be almost entirely a review. However, if you have not had advanced calculus then take some comfort that most of the missing details here are provided in that course. 1.1 points and vectors n n We define R = R × R × · · · × R for n = 1; 2; 3;::: . A point p 2 R has cartesian coordinates p1; p2; : : : ; pn. These are labels, not powers, in our notation. Addition and scalar multiplication are: (p + q)i = pi + qi &(cp)i = cpi n for i 2 Nn. Notice, in the context of R if we take two points p and q then p + q and cp are once n more in R . However, if the space we considered was a solid sphere then you might worry that the n sum of two points landed outside. This space R is essentially our mathematical universe for the most part in this course. We study curves, surfaces and manifolds2 and many of the calculations n we make are reasonable since these curves, surfaces and manifolds are sets of points in R (often n = 3 for this course). 3 j n Define (ei) = δij and note p 2 R is formed by a linear combination of this standard basis: 1 2 n 1 n 1 n p = (p ; p ; : : : ; p ) = p (1; 0;:::; 0) + ··· + p (0;:::; 1) = p e1 + ··· p en: 1pun partly intended 2 n ok, if all goes well, we'll see some examples of manifolds which are not in R , but, for now... 3 in case you forgot, δij = 0 if i 6= j and is 1 if i = j. 5 6 CHAPTER 1. INTRODUCTION Of course, we can either envision p as the point with Cartesian coordinates p1; : : : ; pn , or, we can envision p as a vector eminating from the origin out to the point p. However, this identification n is only reasonably because of the unique role (0; 0;:::; 0) plays in R . When we attach vectors to points other than the origin we really should take care to denote the attachment explicitly. Definition 1.1.1. tangent space and the tangent bundle. n n n We define TpR = fpg × R to be the tangent space to p of R . The tangent bundle of n n n R is defined by T R = [p2Rn fpg × TpR . n Conceptually, TpR is the set of vectors attached or based at p and the tangent bundle is the n collection of all such vectors at all points in R . By a slight abuse of notation, a typical element of n TpR has the form (p; v) where p is the point of attachment or base-point and v is the vector part of (p; v). n The set TpR is naturally a vector space by: (p; v) + (p; w) = (p; v + w)& c(p; v) = (p; cv): Moreover, the dot-product and norm are defined by the usual formulas on the vector part: p (p; v) • (p; w) = v • w & jj (p; v) jj = jjvjj = v • v: n n We say (p; v); (p; w) 2 TpR are orthogonal when (p; v) • (p; w) = v • w = 0. Given S ⊆ TpR ? we may form S = f(p; v) j (p; v) • (p; s) = 0 for all (p; s) 2 Sg. When S is also a subspace of the n ? tangent space at p we have TpR = S ⊕ S . Furthermore, if S plays the role of the tangent space to some object at the point p then S? is identified as the normal space at p. We see the size of the n normal space varies according to the ambient R . For example, a curve C has a one-dimensional 2 tangent space and the normal space has dimension n − 1. In R you have a normal line to C, 3 4 in R there is a normal plane to C, in R there is a normal 3-volume to C. Or, for a surface S 3 with a two-dimensional tangent plane, we have a normal line for S in R , or a normal plane for 4 3 S in R . Much of what is special to R depends directly on the fact that the normal space to a line is a plane and the normal space to a plane is a line. This duality is implicitly used in many steps. n If we assign a vector to each point along S ⊆ R then we say such assignment is a vector field on S. 3 n A vector field on S naturally corresponds to a function X : S ! T R such that X(p) 2 fpg × TpR n n n for each p 2 S. There is a natural mapping π : T R ! R defined by π(p; v) = p for all (p; v) 2 T R . n Notice, to say X : S ! T R is a vector field on S is to say π(X(p)) = p for each p 2 S. Or, simply n π ◦ X = idS. In the usual langauge of bundles we say X is a section of T R over S. Again, keeping with the theme of generality in this section, S could be a curve, surface or higher-dimensional manifold. In each case, we can use a section of the tangent bundle to encapsulate what it means to have a vector field on that object. n n×1 In these notes, for the sake of matrix calculations, I consider R = R (column vectors). This n means we can use standard linear algebra when faced with questions about TpR . We simply remove the p on (p; v) and apply the usual theory. In the next section we find a new notation for (p; v) which brings new insight and is standard in modern treatments of manifold theory. 1.2. ON TANGENT AND COTANGENT SPACES AND BUNDLES 7 1.2 on tangent and cotangent spaces and bundles n Before we define the directional derivative on R we pause to define coordinate functions. Definition 1.2.1. coordinate functions Let xi(p) = pi for i = 1; 2; : : : ; n. In n = 3, x(p) = p1, y(p) = p2 and z(p) = p3.
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