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Notes on Math 321 Notes on Math 321 September 2, 2007 ii °c 2006 Guowu Meng. All rights reserved. Contents Preface v 1 Curves 1 1.1 Basic De¯nitions ...................................... 1 1.2 Frenet Trihedron ...................................... 3 1.3 Local canonical form .................................... 4 1.4 Exercises .......................................... 5 2 Regular Surfaces 7 2.1 Digression: Calculus .................................... 8 2.1.1 Chain Rule ..................................... 8 2.1.2 Inverse Function Theorems ............................ 8 2.2 Basic De¯nitions ...................................... 9 2.2.1 Proof of theorem 2.2.1 ............................... 10 2.2.2 Proof of theorem 2.2.3 ............................... 11 2.3 Existence of tangent planes ................................ 11 2.4 Local Canonical Form ................................... 12 2.5 Exercises .......................................... 13 3 Gauss Map 15 3.1 Smooth maps between regular surface .......................... 15 3.2 Orientation, Gauss map and curvatures ......................... 15 3.3 Digression: Linear Algebra ................................ 17 3.3.1 One-form ...................................... 17 3.3.2 Two-form ...................................... 17 3.3.3 The area form on an oriented Euclidean plane .................. 19 3.4 Fundamental Forms .................................... 20 3.4.1 The total di®erentials ............................... 20 3.4.2 The ¯rst fundamental two-form .......................... 22 3.4.3 The second fundamental two-form ........................ 22 3.4.4 Area Form ..................................... 23 3.5 Exercises .......................................... 23 iii iv CONTENTS 4 Intrinsic Theory 25 4.1 Vector Fields ........................................ 25 4.1.1 Lie derivatives of smooth functions ........................ 26 4.1.2 Gradient vector ¯elds ............................... 27 4.1.3 Lie derivatives of Rm-valued smooth functions ................. 27 4.2 Directional Derivatives ................................... 27 4.2.1 Directional derivatives of Rm-valued smooth functions ............. 27 4.2.2 Covariant derivatives of vector ¯elds ....................... 28 4.2.3 The covariant di®erentiation is non-commutative in general .......... 29 4.3 Riemann Curvature .................................... 30 4.3.1 Digression: Lie bracket ............................... 31 4.3.2 Curvarure Tensor .................................. 31 4.4 The covariant di®erentiation is intrinsic ......................... 32 4.5 Riemann curvature and Gauss curvature are all intrinsic ................ 33 4.6 Exercises .......................................... 33 5 Selected topics 35 5.1 Geodesics .......................................... 35 5.1.1 Digression ...................................... 35 Preface Rough speaking, geometry is the study of interesting shapes and relations among them. (More generally, the core of mathematics is the study of interesting objects and relations among them. The same is true for physics.) Geometry is intuitive, one can quickly get many nice facts by just looking at the geometric shapes | a picture is worth a thousand words. For example, two straight lines on a plane either don't intersect or intersect exactly at one point. Surely, everybody agrees on this simple fact, but most people cannot give a rigorous proof. To give a proof, one needs algebra, and that lead us to analytic geometry. In a sense algebra is a tool which makes our geometric intuition rigorous. Moreover, algebra extends our understanding of geometry beyond our intuition; for example, the fact that one cannot trisect an angle by using a compass and a ruler is not intuitively clear, but can be proved by using algebra. For the study of more sophisticated shapes, algebra is not enough, we need a great tool, namely calculus, and that leads us to di®erential geometry. For example, what is the volume of a ball with radius r? Algebra alone cannot give you the answer, but calculus surely can. Roughly speaking, di®erential geometry is the study of interesting shapes and relations among them by using calculus. But I prefer to view di®erential geometry as college calculus III | calculus on curved spaces. To me, there is a huge psychological bene¯t to adopt this view: di®erential geometry is just a further extension of our regular calculus, so we are just repeating many of the same old calculus ideas most of the time during the learning process; in other words, we are fearless while wandering because we are not far away from our \mother" (i.e., the regular calculus). In this course, we only study curves and surfaces inside R3; these are nice geometric shapes of dimension one and two respectively. Something new such as curvatures and torsion appear naturally, they tell us how the the curves and surfaces sit inside R3 up to the rigid motions of R3 (i.e., the transformations of R3 which preserve both lengths and angles.) These facts are quite intuitive. While the curves are intrinsically flat (i.e., can be straightened locally without stretching); for surface, much less intuitive facts were discovered by Gauss: 1) the Gauss curvature K is intrinsic, i.e., stays the same no matter how you bend the surface without stretching, 2) the total Gauss curvature of an (oriented) closed surface is topological, i.e., stays the same no matter how you stretch the surface. It is worth to mention that the Gauss curvature K also has a meaning in calculus! Recall that, in regular calculus of two variables, for smooth vector ¯eld F ,(@x@y ¡ @y@x)F is zero. If we extend the calculus to a surface, then (@x@y ¡ @y@x)F = KF , so the Gauss curvature measures the the failure of commutativity for mixed partial derivatives. This observation is quite natural and fruitful; as a matter of fact, if we emphasize the view that di®erential geometry is the calculus on v vi PREFACE curved spaces, we could rediscover pretty much all mathematics that we need for writing down the fundamental laws of physics: from Riemannian geometry to geometry of vector bundles. Chapter 1 Curves Prerequisite: Calculus I, i.e., calculus for vector-valued functions of one variable | the kind of calculus needed for the study of the Newtonian mechanics. Curves are one dimensional nice geometric objects, and it is very useful to view it as the trajectory of a moving particle in R3 or R2. 1.1 Basic De¯nitions Some standard notations: ² I | an open interval, say (¡1; 2) or (¡1; 1) or (2; 1), etc.; ² t0 | a number in I; ² f | a function on I taking values in R or R2 or R3; ² ® | a parameterized smooth curve; ² C | a regular curve. Some standard terminologies: ² f is smooth | each component function of f is a smooth (i.e., in¯nitely di®erentiable) real-valued function; De¯nition 1.1.1. A parameterized smooth curve in R3 is just a smooth map ®: I / R3. The image of ® (viewed as a map) is called the trace of ® or a curve. We say ® is regular if ®0 is nonzero everywhere on I.A regular curve in R3 is just the trace of a regular parameterized smooth curve in R3. In other word, a regular curve in R3 is just a subset of R3 which can be parameterized by a smooth map ® with nowhere vanishing ¯rst derivative. By the way, curves in R2, i.e., plane curves, can be similarly de¯ned. 1 2 CHAPTER 1. CURVES Example 1.1.1. ®(t) = (t; t) is a regular parameterized smooth curve in R2, and its trace ( a straight line) is a regular plane curve. Example 1.1.2. ®(t) = (t3; t3) is a parameterized smooth curve in R2, but not regular at t = 0. However, its trace ( a straight line) is a regular plane curve. So a regular curve can have a non-regular parametrization. Example 1.1.3. ®(t) = (t3; t2) is a parameterized smooth curve in R2, but it is not a regular parameterized smooth curve. One can show that its trace is not a regular curve1. So a non-regular curve can have a smooth parametrization. Example 1.1.4. ®(t) = (t; jtj) is not a parameterized smooth curve in R2 because it is not smooth at t = 0. One can show that its trace is not a regular curve either2. Example 1.1.5. Let f be a real valued smooth function on I. Then the graph of f: ¡(f) := f(t; f(t) j t 2 Ig is a regular plane curve. Theorem 1.1.1 (Arclength Parametrization). Let C be a regular curve, then C admits an arclength parametrization, i.e., C is the trace of a regular parameterized smooth curve ® such that j®0j = 1 everywhere. / 3 Proof. Since C is a regular curve, it must be theR trace of a parameterized smooth curve ¯: J R such that j¯0j 6= 0 everywhere. Let s = g(t) = t j¯0(x)j dx. Since3 g0 = j¯0j > 0 on J, g is smooth t0 and has a smooth inverse t = f(s) with 1 1 f 0(s) = = : g0(f(s)) j¯0(f(s))j Let ®(s) = ¯(f(s)) for s 2 I (where I is the image of f). Then j®0(s)j = j¯0(f(s))j ¢ f 0(s) = 1. Example 1.1.6. The unit circle centered at (0; 0) is a regular plane curve. ®(s) = (cos t; sin t) is one of its arclength parameterizations. 3 De¯nition 1.1.2. Suppose that ®: I / R is a regular parameterized smooth curve and t0 2 I. The trace of the linear approximation of ® around t0 is called the tangent line of ® at t0. Note that the tangent line of ® at t0 is the trace of this parameterized curve: 0 t 7! ®(t0) + ® (t0)(t ¡ t0): 1The proof is given in example 1.3.1. 2I leave its proof to you. 3I will prove this statement in class. 1.2. FRENET TRIHEDRON 3 1.2 Frenet Trihedron Suppose that ®: I / R3 is a regular parameterized smooth curve. By a re-parametrization if necessary, we can assume that j®0j = 1 on I. Furthermore, we assume that j®00j 6= 0 on I. Let ~t = ®0 and we call ~t(s) the tangent vector at s.
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