# Riemannian geometry a beginners guide - Morgan F.

ISBN 0-86720-242-4

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^sin — tjW, which vanishes at the endpoints of y. By (4), the initial second variation of length is given by

(■

■)

L"(0) = I f-cos-r) - (sin2-t]K(T,W)

J \l I ) \ I )

0

" TT2 2 7T

< — cos —t sin —t = 0.

. I2 ll2 I

0

88 CHAPTER 9

The Rauch Comparison Theorem. One of the main ingredients in a proof, and one of the most useful tools in Riemannian geometry, is the Rauch Comparison Theorem. It says, for example, the following:

Let Mi, M2 be complete smooth Riemannian manifolds with sectional curvatures Kx^ K0^ K2 for some constant K0. For px E Mx, p2 E M2, identify T = TP1Mi = TP2M2 via a linear isometry. Let B be an open ball about 0 in T on which Exppl and ExpP2 are diffeomor-phisms into Mx and M2. Let y be a curve in B, and let yx, y2 be its images in Mx, M2. Then length (yx) ^ length (y2).

In applications, either Mx or M2 is usually taken to be a sphere, Euclidean space, or hyperbolic space, all of which have well-known trigonometries. Thus one obtains distance estimates on the other manifold from curvature bounds.

10

General Norms

In nature, the energy of a path or surface often depends on direction as well as length or area. The surface energy of a crystal, for example, depends radically on direction. Indeed, some directions are so much cheaper that most crystals use only a few cheap directions. (See Figure 10.1.) This chapter applies more general costs or norms <I> to curves and presents an appropriate generalization of curvature.

10.1 Norms. A norm Ф on R” is a nonnegative, convex homogeneous function on R". We call Ф Ck if its restriction to R" - {0} is Ck (or, equivalently, if its restriction to the unit sphere S'1-1 is Ck). The convexity of Ф is equivalent to the convexity of its unit ball

{*: Ф(*) < 1}.

For any curve C, parametrized by a differentiable map у: [0,1] —» R", define

Ф(С) = I Ф(Т) ds = I Ф(у) dt. с [0,1]

If С is a straight line segment, then

Ф(С) = Ф(Т) length C.

GENERAL NORMS 91

/

/

Figure 10.2. Since the unit ball of <1> is strictly convex, there is a linear function or 1-form <p such that <p(u) < ^(u), with equality only if u = B — A.

10.2. Proposition. Among all differentiable curves С from A to В, the straight line L minimizes Ф(С) uniquely if Ф is strictly convex.

Proof. Since the unit ball of Ф is convex, there is a constant-coefficient differential form <p such that

(p(v) < Ф(и),

with equality when v = B — A. (See Figure 10.2.) If Ф is strictly

Figure 10.1. Crystal shapes typically have finitely many flat facets corresponding to surface orientation of low energy. (The first two photographs are from Steve Smale’s Beautiful Crystals Calendar; current version available for $12 from 69 Highgate Road, Kensington, CA 94707. The third photograph is from E. Brieskorn. All three appeared in Mathematics and Optimal Form by S. Hildebrandt and A. Tromba [HT. p. 181].)

92 CHAPTER 10

convex, equality holds only if v is a multiple of B — A. Let C' be any differentiable curve from A to B. Then

by Stokes’s Theorem, so C is ^-minimizing. If <I> is strictly convex, the inequality is strict unless C' is also a straight line from A to B, so C is uniquely minimizing.

10.3. Proposition. A nonnegative homogeneous C2 function on R" is convex (respectively, uniformly convex) if and only if the restrictions d>(0) of to circles about the origin satisfy

Proof. Since convexity in every plane through 0 is equivalent to convexity, we may assume n = 2. The curvature k of any graph r =/(0) in polar conditions is given by

Therefore the curvature of the boundary of the unit ball r = l/<b(0) is given by

The proposition follows.

10.4. Generalized curvature. Let C be a C2 curve with arc length parametrization f: [0,1] -t»R" and curvature vector k. Let <£> be a C2 norm. Consider variations 8f supported in (0,1). Then the first variation satisfies

| q>ds = <$>(C)

c

<I>"(0) + 0(0) < 0 (<0).

f2 - //" + 2/'2

(/2 +/,2)3/2

GENERAL NORMS 93

where D2<t> represents the second derivative matrix evaluated at the unit tangent vector. In particular, for the case of length (<£(.*) = L(x) = |*|),

8L(f)=- f k -8fds.

[0,1]

In general, we call D2<£>(k) the generalized 4>-curvature vector.

Proof. Since <£(/) = / <*>(/'(«)) du for any parameterization

/(«)>

«<&(/)= f D<t>(f')-Sf'(u)du

= - jV<I>(/’)(/")• V(u)du

by integration by parts. Since for the initial arc length parametriz-ation, /' is the unit tangent vector and /" is the curvature vector k, initially

«*(/) = - [D24-(K)-5/(i)dS.

10.5. The isoperimetric problem. One famous isoperimetric theorem says that among all closed curves C in R" of fixed length, the circle bounds the most area—that is, the oriented area-minimizing surface S of greatest area (see, for example, [F, 4.5.14]). In other words, an area-minimizing surface S with given boundary C satisfies

area S ^ — (length C)2.

477

Given a convex norm or R", we seek a closed curve C0 of fixed cost (C0) which bounds the most area, so any area-minimizing surface S with given boundary C satisfies

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