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Riemannian geometry a beginners guide - Morgan F.

Morgan F. Riemannian geometry a beginners guide - Jones and Bartlett, 1993. - 121 p.
ISBN 0-86720-242-4
Download (direct link): riemanniangeometry1993.djvu
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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 )
" TT2 2 7T
< — cos —t sin —t = 0.
. I2 ll2 I
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.
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.
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 $12 from 69 Highgate Road, Kensington, CA 94707. The third photograph is from E. Brieskorn. All three appeared in Mathematics and Optimal Form by S. Hildebrandt and A. Tromba [HT. p. 181].)
convex, equality holds only if v is a multiple of B — A. Let C' be any differentiable curve from A to B. Then
by Stokes’s Theorem, so C is ^-minimizing. If <I> is strictly convex, the inequality is strict unless C' is also a straight line from A to B, so C is uniquely minimizing.
10.3. Proposition. A nonnegative homogeneous C2 function on R" is convex (respectively, uniformly convex) if and only if the restrictions d>(0) of to circles about the origin satisfy
Proof. Since convexity in every plane through 0 is equivalent to convexity, we may assume n = 2. The curvature k of any graph r =/(0) in polar conditions is given by
Therefore the curvature of the boundary of the unit ball r = l/<b(0) is given by
The proposition follows.
10.4. Generalized curvature. Let C be a C2 curve with arc length parametrization f: [0,1] -t»R" and curvature vector k. Let <£> be a C2 norm. Consider variations 8f supported in (0,1). Then the first variation satisfies
| q>ds = <$>(C)
<I>"(0) + 0(0) < 0 (<0).
f2 - //" + 2/'2
(/2 +/,2)3/2
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.
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.
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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