Lecture Notes for 120 - UCLA Department of Mathematics

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6.2. GENERALIZED AND ABSTRACT SURFACES 138 (6) (Liouville surfaces) The first fundamental form is given by apple r 2 0 [I] = 0 r 2 where r 2 = f (u)+g (v) > 0. (7) (Monge patch) The first fundamental form is given by apple 1+f 2 fg [I] = fg 1+g 2 where f = @F @F @u ,g = @v and F = F (u, v). 6.2. Generalized and Abstract Surfaces It is possible to work with generalized surfaces in Euclidean spaces of arbitrary dimension: q (u, v) :U ! R k for any k 2. What changes is that we no longer have asinglenormalvectorN. In fact for k 4 there will be a whole family of normal vectors, not unlike what happened for space curves. What all of these surfaces do have in common is that we can define the first fundamental form. Thus we can also calculate the Christoffel symbols of the first and second kind using the formulas in terms of derivatives of g. This leads us to the possibility of an abstract definition of a surface that is independent of a particular map into some coordinate space R k . One of the simplest examples of a generalized surface is the flat torus in R 4 . It is parametrized by q (u, v) = (cos u, sin u, cos v, sin v) and its first fundamental form is I= apple 1 0 0 1 So this yields a Cartesian parametrization of the entire torus. This is why it is called the flat torus. It is in fact not possible to have a flat torus in R 3 . An abstract parametrized surface consists of a domain U ⇢ R 2 and a first fundamental form [I] = apple guu g uv where g uu , g vv , and g uv are functions on U. The inner product of vectors X = (X u ,X v ) and Y =(Y u ,Y v ) thought of as having the same base point p 2 U is defined as I(X, Y )= ⇥ X u X ⇤ apple apple v g uu (p) g uv (p) Y u g vu (p) g vv (p) Y v For this to give us an inner product we also have to make sure that it is positive definite: 0 < I(X, X) g uv g vv = ⇥ X u X ⇤ apple apple v g uu g uv X u g uv g vv X v = X u X u g uu +2X u X v g uv + X v X v g vv Proposition 6.2.1. I is positive definite if and only if g uu + g vv , and g uu g vv (g uv ) 2 are positive.
6.2. GENERALIZED AND ABSTRACT SURFACES 139 Proof. If I is positive definite, then we see that g uu and g vv are positive by letting X =(1, 0) and (0, 1). NexttakeX = p g vv , ± p g uu to get Thus showing that 0 < I(X, X) =2g uu g vv ± 2 p g uu p gvv g uv ±g uv (g uv ) 2 . To check that I is positive definite when g uu + g vv , and g uu g vv positive we start by noting that g uu g vv >g 2 uv 0 (g uv ) 2 are Thus g uu and g vv have the same sign. As their sum is positive we have that both of the terms are positive. It then follows that I(X, X) = X u X u g uu +2X u X v g uv + X v X v g vv X u X u g uu = (|X u | p g uu |X v | p g vv ) 2 0 2 |X u ||X v | p g uu g vv + X v X v g vv Here is first inequality is in fact > unless X u =0or X v =0. When, say, X u =0 we obtain I(X, X) =(X v ) 2 g vv > 0 unless also X v =0. There is an interesting example of an abstract surface on the upper half plane H = {(u, v) | v>0} , where the metric coefficients are [I] = apple guu g uv = 1 apple 1 0 g vu g vv v 2 0 1 One can show that for each point p 2 H, there is a small neighborhood U containing p and q (u, v) :U ! R 3 such that ⇥ @q @u @q @v ⇤ t ⇥ @q @u @q @v ⇤ = apple @q @u · @q @u @q @v · @q @u @q @u · @q @v @q @v · @q @v = 1 v 2 apple 1 0 0 1 In other words we can locally represent the abstract surface as a surface in space. However, a very difficult theorem of Hilbert shows that one cannot represent the entire surface in space, i.e., there is no parametrization q (u, v) :H ! R 3 defined on the entire domain such that ⇥ @q @u @q @v ⇤ t ⇥ @q @u @q @v ⇤ = apple @q @u · @q @u @q @v · @q @u @q @u · @q @v @q @v · @q @v = 1 v 2 apple 1 0 0 1 Janet-Burstin-Cartan showed that if the metric coefficients of an abstract surface are analytic, then one can always locally represent the abstract surface in R 3 . Nash showed that any abstract surface can be represented by a map q (u, v) :U ! R k on the entire domain, but only at the expense of making k very large. Based in part on Nash’s work Greene and Gromov both showed that one can always locally represent an abstract surface in R 5 . ⇤
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Classical Differential Geometry Pet
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CONTENTS 4 Chapter 7. Other Topics
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1.1. CURVES 6 measurements are need
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1.1. CURVES 8 as we then get q (✓
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1.1. CURVES 10 these derivatives ar
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1.1. CURVES 12 (a) Show that if r (
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1.2. ARCLENGTH AND LINEAR MOTION 14
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1.3. CURVATURE 20 relates the side
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1.3. CURVATURE 22 of a that is norm
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1.3. CURVATURE 24 As the piece of t
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1.4. INTEGRAL CURVES 26 (14) Let q
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1.4. INTEGRAL CURVES 28 So at q 0 =
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CHAPTER 2 Planar Curves 2.1. Genera
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2.2. THE FUNDAMENTAL EQUATIONS 32 M
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2.2. THE FUNDAMENTAL EQUATIONS 34 (
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2.3. LENGTH AND AREA 36 (22) (Newto
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As ( 2.3. LENGTH AND AREA 38 sin
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2.3. LENGTH AND AREA 40 The choice
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2.4. THE ROTATION INDEX 42 the hypo
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2.4. THE ROTATION INDEX 44 where an
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2.4. THE ROTATION INDEX 46 Exercise
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2.5. TWO SURPRISING RESULTS 48 unit
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2.6. CONVEX CURVES 50 Exercises. Po
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2.6. CONVEX CURVES 52 Proof. Any cu
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2.6. CONVEX CURVES 54 (9) Give an e
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3.1. THE FUNDAMENTAL EQUATIONS 56 T
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3.1. THE FUNDAMENTAL EQUATIONS 58 B
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3.1. THE FUNDAMENTAL EQUATIONS 60 (
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.3. CLOSED SPACE CURVES 66 (12) Pr
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3.3. CLOSED SPACE CURVES 68 Thus th
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3.3. CLOSED SPACE CURVES 70 (b) Sho
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CHAPTER 4 Basic Surface Theory 4.1.
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4.1. SURFACES 74 Definition 4.1.6.
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4.1. SURFACES 76 (6) Many classical
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4.2. TANGENT SPACES AND MAPS 78 Thi
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4.2. TANGENT SPACES AND MAPS 80 We
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4.2. TANGENT SPACES AND MAPS 82 Def
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4.3. THE ABSTRACT FRAMEWORK 84 Befo
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4.3. THE ABSTRACT FRAMEWORK 86 in o
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B.6. CHEBYSHEV NETS 188 (1) Show th
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B.7. ISOTHERMAL COORDINATES 190 K =
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Lecture Notes for 120 - UCLA Department of Mathematics

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