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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 Previous << 1 .. 13 14 15 16 17 18 < 19 > 20 21 22 23 24 .. 25 >> Next ^sin ŌĆö tjW, which vanishes at the endpoints of y. By (4), the initial second variation of length is given by
(Ō¢Ā
Ō¢Ā)
L"(0) = I f-cos-r) - (sin2-t]K(T,W)
J \l I ) \ I )
0
" TT2 2 7T
< ŌĆö cos ŌĆöt sin ŌĆöt = 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 ąż on RŌĆØ is a nonnegative, convex homogeneous function on R". We call ąż 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 ąż is equivalent to the convexity of its unit ball
{*: ąż(*) < 1}.
For any curve C, parametrized by a differentiable map čā: [0,1] ŌĆö┬╗ R", define
ąż(ąĪ) = I ąż(ąó) ds = I ąż(čā) dt. čü [0,1]
If ąĪ is a straight line segment, then
ąż(ąĪ) = ąż(ąó) 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 ŌĆö A.
10.2. Proposition. Among all differentiable curves ąĪ from A to ąÆ, the straight line L minimizes ąż(ąĪ) uniquely if ąż is strictly convex.
Proof. Since the unit ball of ąż is convex, there is a constant-coefficient differential form <p such that
(p(v) < ąż(ąĖ),
with equality when v = B ŌĆö A. (See Figure 10.2.) If ąż 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 SmaleŌĆÖs 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 ŌĆö A. Let C' be any differentiable curve from A to B. Then
by StokesŌĆÖs 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 <┬Ż> 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 (<┬Ż(.*) = L(x) = |*|),
8L(f)=- f k -8fds.
[0,1]
In general, we call D2<┬Ż>(k) the generalized 4>-curvature vector.
Proof. Since <┬Ż(/) = / <*>(/'(┬½)) du for any parameterization
/(┬½)>
┬½<&(/)= f D<t>(f')-Sf'(u)du
= - jV<I>(/ŌĆÖ)(/")ŌĆó 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
┬½*(/) = - [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 areaŌĆöthat 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 ^ ŌĆö (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 Previous << 1 .. 13 14 15 16 17 18 < 19 > 20 21 22 23 24 .. 25 >> Next 