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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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The unit tangent vector T = v/|v| = v/c. Hence the curvature
10 CHAPTER 2 vector is
k = dT Ids = ^ = — = — {—a cos t, —a sin t, 0),
dTldt _ v/c _ 1 dsl dt c c2
and the scalar curvature is k = ale2. Therefore the ideal inner radius is
r = 1 Ik = c2la ~ 10.45 feet, in close agreement with the engineer’s experiment.
Surfaces in R
This chapter studies the curvature of a C2 surface S C R3 at a point p E S. (C2 just means that locally, the surface is the graph of a function with continuous second derivatives. Computing the curvature will involve differentiating twice.) Let TPS denote the tangent space of vectors tangent to S at p. Let n denote a unit normal to S at p. To study the curvature of S, we slice S by planes containing n and consider the curvature vector k of the resulting curves. (See Figure
3.1.) Of course each such k must be a multiple of n: k = ku. (For now we will allow k to be positive or negative. The sign of k depends on the choice of unit normal n.) It will turn out that the largest and the smallest curvatures k1? k2 (called the principal curvatures) occur in orthogonal directions and determine the curvatures in all other directions.
Choose orthonormal coordinates on R3 with the origin at p, S tangent to the jc, y-plane at p, and n pointing along the positive z-axis. Locally S is the graph of a function z = /(*, y). Any unit vector v tangent to S at p, together with the unit normal vector n, spans a plane, which intersects S in a curve. The curvature k of this curve, which we call the curvature in the direction v, is just the second derivative
2 O’) ^ (,)
dx dy
_dxdy dy (p) _
Figure 3.1. The curvature of a surface S at a point p is measured by the curvature of its slices by planes.
For example, if
The bilinear form (D2f)p on TPS is called the second fundamental form II of S at p, given in coordinates as a symmetric 2x2 matrix:
II = D2f =
This formula is good only at the point where the surface is tangent to the Jt,y-plane. For the second fundamental form, we will always use orthonormal coordinates.
Since II is symmetric, we may choose coordinates x, y such that II is diagonal:
" aV a2/'
dx2 dx dy
d2f aV
_ dx dy dy2 _
L 0 k2\
Then the curvature k in the direction v = (cos 6, sin 6) is given by Euler’s formula (1760):
k = II(v, v) = vrIIv = Ki cos2 6 + k2 sin2 6,
a weighted average of and k2. In particular, the largest and smallest curvatures are ki and k2, obtained in the orthogonal directions we have chosen for the x- and y-axes.
3.1. Definitions. At a point p in a surface S C R3, the eigenvalues Ku k2 of the second fundamental form II are called the principal curvatures, and the corresponding eigenvectors (uniquely determined unless Ki = k2) are called the principal directions or directions of curvature. The trace of II, kx + k2, is called the mean curvature H. The determinant of II, kik2, is called the Gauss curvature G.
Note that the signs of II and H but not of G depend on the choice of unit normal n. Some treatments define the mean curvature as
itraceII = ^i±^.
2 2
Just as the curvature k gave the rate of change of the length of an evolving curve in Chapter 2, the mean curvature H gives the rate of change of the area of an evolving surface. Just as the rate of change of a function of several variables is called the directional derivative and depends on the direction of change, the initial rate of change of the area of a surface depends on its initial velocity V and is called the first variation.
3.2. Theorem. Let S be a C2 surface in R3. The first variation of the area of S with respect to a compactly supported C2 vectorfield V of S is given by integrating Y against the mean curvature:
8\S) = - J V • Hn.
Remark. S1(S) is defined as
— area (5 + t\ )|f=0, dt
or, equivalently,
-j area (/,(5))|,=0, at
where ft is any C3 deformation of space with initial velocity V on S. [51(5) depends only on Y and is linear in V.] If S has infinite area, restrict attention to spt V.
Proof. Since the formula is linear in V, we may consider tangential and normal variations separately. For tangential variations, which correspond to sliding the surface along itself, 51(5) = 0, confirming the formula. Let Vn be a small normal variation, and consider an infinitesimal square area dxdy at p, where we may assume the principal directions point along the axes. To first order, the new infinitesimal area is
(1 - Vkj) dx(l - Vk2) dy~{ 1 - VH)dx dy = (1 - V • Hn)dx dy
(compare to Figure 2.2). The formula follows.
Remark. A physical surface such as a soap film would tend to move in the normal direction of positive mean curvature in order to decrease its area, unless balanced by an opposite pressure. The mean curvature of a soap bubble in equilibrium is proportional to the pressure difference across it.
3.3. Minimal surfaces. It follows from Theorem 3.2 that an area-minimizing surface, which minimizes area in competition with surfaces with the same boundary, must have vanishing mean curvature. Any surface with 0 mean curvature is called a minimal surface.
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