Differential Geometry @

Differential Geometry @ UCI Math 162AB Author: Zhiqin Lu Institute: University of California, Irvine Date: September 15, 2020 Version: 2.0 Bio: Zhiqin Lu is a professor at UCI Lecture Notes based on Barrett O’Neill’s book Elementary Differential Geometry. Partially supported by a NSF grant DMS-19-08513. Contents 0 Preliminaries1 0.1 Vector Spaces . .1 0.2 Inner Product . .3 0.3 Linear Transformation . .6 0.4 Lines, planes, and spheres . .9 0.5 Vector Calculus . 10 0.6 Einstein Convention . 12 1 Calculus on Euclidean Space 15 1.1 Euclidean Space . 15 1.2 Tangent Vectors . 16 1.3 Directional Derivatives . 17 3 1.4 Curves in R ................................... 19 1.5 1-forms ...................................... 22 1.6 Differential Forms . 24 1.7 Mappings . 27 2 Frame Fields 31 2.1 Dot Product . 31 2.2 Curves . 33 2.3 The Frenet Formulas . 35 2.4 Arbitrary-Speed Curves . 39 2.5 Covariant Derivatives . 45 2.6 Frame Fields . 46 2.7 Connection forms . 49 4 Calculus on Surface 18 Chapter 4 Exercise . 20 4.1 .......................................... 20 4.2 .......................................... 20 4.3 .......................................... 20 4.4 .......................................... 20 4.5 .......................................... 20 4.6 Integration of Forms . 20 4.7 Topological properties of surfaces . 22 4.8 Manifolds . 24 CONTENTS 5 Shape Operators 26 3 5.1 The shape operator of M ⊂ R .......................... 26 5.2 Normal Curvature . 27 5.3 Gauss Curvature . 28 5.4 Computational Techniques . 30 5.5 The implicit Case . 31 5.6 Surface of revolution . 33 3 6 Geometry of Surfaces in R 36 6.1 The fundamental equations . 36 6.2 Isometries and local isometries . 39 6.3 Intrinsic geometry of surfaces in . 39 6.4 Orthogonal Coordinates . 40 6.5 Integration and Orientation . 40 6.6 Total Curvature . 41 6.7 Congruence of surfaces . 42 ii Chapter 0 Preliminaries Introduction h Vector space h Lines, planes, and spheres h Inner product on vector space h Einstein Convention h Linear transformation h Vector Calculus 0.1 Vector Spaces Definition 0.1. Vector Space A vector space V is a nonempty set with two binary operations “+” and scalar multiplication “·” satisfying the following eight axioms: let u; v; w 2 V and r; s 2 R, we have u + v = v + u; u + (v + w) = (u + v) + w; 0 + u = u + 0 = u for a vector called 0; (rs) · u = r · (s · u); (r + s) · u = r · u + s · u; r · (u + v) = r · u + r · v; 0 · u = 0; 1 · u = u. | As is well-known, the set V satisfying the first three axioms form an Abel semi-group. The existence of the inverse of a vector u can be verified by using the scalar multiplication. Let u be a vector, we claim that (−1) · u is the inverse of u because u + (−1) · u = (1 + (−1)) · u = 0: Similarly, we can define the subtraction by u − v = u + (−1) · v: Note We sometimes omit the · and write, for example, ru for r · u. External Link. Here is the video explanation of vector space (linear indepedence). Useful! External Link. Here is the Math 162A pre-requisite videos. In linear algebra, we restrict ourselves to finite dimensional vector space. But, a lot of results in finite dimensional case can be extended to infinite dimensional case as well as abstract vector space cases. 0.1 Vector Spaces In the following, we give some examples of vector spaces. n Example 0.1 R , the n-dimensional Euclidean space, is the set of all n-vectors 8 2 3 9 x > 1 > n < 6 . 7 = R = x = 6 . 7 j xi 2 R : 4 5 > > : xn ; Example 0.2 The set of all m × n matrices for an (mn)-dimensional vector space. Example 0.3 The space of polynomials of degree no more than n, where n is a nonnegative integer, is a vector space. A single-variable polynomial of degree no more than n can be expressed as n p(t) = a0 + a1t + ··· + ant : Let n q(t) = b0 + b1t + ··· + bnt be another polynomial. Define the addition to be n (p + q)(t) = (a0 + b0) + (a1 + b1)t + ··· + (an + bn)t ; and the scalar multipliction to be: let λ 2 R n (λp)(t) = (λa0) + (λa1)t + ··· + (λan)t : With respect to the addition and scalar multiplication, the space is a vector space. In the above two examples, the dimensions of the vector spaces are finite. Let’s show some examples of infinite dimensional vector spaces. Note The set of all polynomials of degree equal to n is not a vector space. Example 0.4 Moreover, the space of all real-valued functions on a set is a (infinite dimensional) vector space. Let’s recall the definition of a function. Definition 0.2 A function f : X ! Y is a triple (f; X; Y ), where X; Y are sets, and f is an assignment, or a rule, that for any element x in X, there is a unique y = f(x) in Y attached to it. X is called the domain of f, and Y is called the codomain of f. The range is the subset of the codomain Y consists of all f(x) when X is running through X. The assignment sometimes is written as x 7! f(x). Thus a complete description of a function can be given as f : X ! Y; x 7! f(x): u v | 2 0.2 Inner Product As above, we can define the addition and scalar multiplication as (f + g)(x) = f(x) + g(x); (λf)(x) = λf(x): Remark The vector space we study in differential geometry are the “abstract” vector space, which is on the contrary to the vector spaces we studied in Math 3A. n The following concepts are defined in abstract vector spaces similar to those in R . 1. linear combination, span; 2. linear dependence and independence; 3. basis and dimension. Remark In infinite dimensional space, a basis is defined by a set of vectors I = fv1; v2; ··· ; vn; · · · g satisfying the following 1. any finite subset of I is linearly independent; 2. any element can be expressed as a (finite) linear combination of element in I. Let (V; +; ·) be a vector space. We can endow geometric structure onto it by defining the concept of inner product. Remark Let S = fv1; v2; ··· ; vng be a finite set. Let V = Span S = Span fv1; v2; ··· ; vng: Then we say S spans V , and S is a spanning set of V . 0.2 Inner Product The addition and scalar multiplication define the algebraic structure of a vector space. In order to introduce geometry to linear algebra, we can endow geometric structure onto it by defining the concept called inner product. Definition 0.3 An inner product on a vector space V is a function h ; i V × V ! R satisfying the following properties: let u; v; w 2 V and r; s 2 R, we have (a): hu; vi = hv; ui Symmetry (b): hu; rv + swi = rhu; vi + shu; wi Linearity (c):hu; ui ≥ 0 and equality is true if and only if u = 0 P ositivity | Once we introduce the inner product, we introduce geometry into vector space. For example 3 0.2 Inner Product Definition 0.4 (Length of a vector) We can define the length, or the norm, of a vector to be kuk = phu; ui: (Distance) Let u; v be two vectors. Then their distance is defined to be dist(u; v) = ku − vk = phu − v; u − vi: | Note in the definition of distance, we used both the geometric structure (inner product) and algebraic structrue (subtraction is defined by u − v = u + (−1) · v). 3 n Example 0.5 In R (and in R ), the ordinary dot product h(a1; a2; a3); (b1; b2; b3)i = a1b1 + a2b2 + a3b3 is an inner product. 3 However, there are many other ways one can define inner product on R . For example, we can define h(a1; a2; a3); (b1; b2; b3)i = 2a1b1 + 3a2b2 + 4a3b3: It is an inner product. In general, let 2 3 r11 r12 r13 6 7 R = 6r r r 7 4 21 22 235 r31 r32 r33 be a positive definite matrix. Then 3 1 2 3 1 2 3 X i j h(a ; a ; a ); (b ; b ; b )i = rija b i;j=1 is an inner product. Example 0.6 We let R[t] be the vector space of all polynomials. Define the inner product Z 1 hp; qi = p(x)q(x) dx −1 More general, if ρ(x) > 0 be a positive continuous function, then Z 1 hp; qi = p(x)q(x)ρ(x) dx −1 is an inner product. One of the most important inequality in mathematics is called the Cauchy-Schwarz Inequal- ity. Theorem 0.1 Let u; v be two vectors. Then we have jhu; vij ≤ kuk · kvk; and the equality holds if and only if u; v are linearly dependent. ~ 4 0.3 Linear Transformation Proof. This is the standard proof. Let t be a real number. Then by the positivity of the inner product, we have hu + tv; u + tvi ≥ 0: Using the linearity, we get t2hv; vi + 2thu; vi + hu; ui ≥ 0: Since the above inequality is true for any real number t, the discriminant ∆ = 4jhu; vij2 − 4hu; ui · hv; vi ≤ 0; which proves the inequality. Proof. [Second Proof] Assume that v 6= 0. Then we have hu; vi hu; vi hu − v; u − vi ≥ 0: hv; vi hv; vi Expanding the above inequality, we obtain the Cauchy-Schwarz Inequality.

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