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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 .. 2 3 4 5 6 < 7 > 8 9 10 11 12 13 .. 25 >> Next n
H= ‚Äî 1 Bni/dXi = ‚Äîdivn.
If the hypersurface is given as a level set {/(*i, ...,*‚Äû) = c}, then n = V/71 V/'l, where V/= (dni/dxi,. . . , dnn/dxn), and
H = -div
It
|V/|
(1)
5.1. Theorem. Let S be a C2 m-dimensional surface in R". The first variation of the area of S with respect to a compactly supported C2 vectorfield V on S is given by integrating V against the mean curvature vector:
8\S) = - | V ‚Ä¢ H.
s
Proof. Since the formula is linear in v, we may consider variations in the Xi,x2,. .. ,xn directions separately. For the x1?.. . ,xm directions, which correspond to sliding the surface along itself, 5X(5) = 0, as the formula says. Let V be a small variation in the x, direction (m <; ^ n), and consider an infinitesimal area dxi ‚Ä¢ ‚Ä¢ ‚Ä¢ dxm at p, where we may assume that the x;- component of II is diagonal:
m-DIMENSIONAL SURFACES IN R" 33
Kl
0
To first order, it is displaced to an infinitesimal area (1 + | V|#d) dxx ‚Ä¢ ‚Ä¢ ‚Ä¢ (1 + | V|kw) dxm ~ (1 - V ‚Ä¢ H)dxx The formula follows.
dxr
5.2. Sectional and Riemannian curvature. The sectional curvature –ö of S at p assigns to every 2-plane P –° TPS the Gauss curvature of the 2-dimensional surface
5–ü(–Ý0–¢—Ä5—Ö).
If v, w give an orthonormal basis for P, then by its definition the sectional curvature is
K(P) = II(i>, u) ‚Ä¢ ll(w, w) - II(u, w) ‚Ä¢ II(d, w). (1)
For example, if II = [ai}] and P = ex –ª e2 is the ^i,jr2-plane, then the sectional curvature is
K(P) = ll(eu ex) ‚Ä¢ II(e2, e2) - ll(eu e2) ‚Ä¢ II(eb e2)
= flu ‚Ä¢ a22 ‚Äî aX2 ‚Ä¢ aX2.
Remark. For hypersurfaces (n = m + 1), for any 2-plane P = 2 Pij e, a e}, if we choose coordinates to make the second fundamental form diagonal,
11 =
kx
0
then
K(P) = 2 p1jk,Kj.
1
Thus any sectional curvature K(P) is a weighted average of the sectional curvatures k, k; of the axis 2-planes et a er
For 2 < m < n, R" = TPS x Rj x ‚Ä¢ ‚Ä¢ ‚Ä¢ x R‚Äû-m, let 5, denote the projection of S into TPS x Rf, with sectional curvature Kt. Then, by (1), the sectional curvature K of S satisfies K = 2 Kt.
34 CHAPTER 5
Hence the sectional curvature of an m-dimensional surface S in R" may be computed by separately diagonalizing the n - m components of II, taking the appropriate weighted average of products of principal curvatures for each component, and summing over all components.
If m = n - 1, then II is a symmetric bilinear form called the second fundamental form. Its eigenvalues kx, . . . , Km are called the principal curvatures. Since (D2f)p is symmetric, in some orthogonal coordinates it is diagonal and / takes the form
Ki*i | k2x\ |
KmXm , 2\
H--------1- o(x ).
In general, if II = (fl/y), then formula (1) yields
k(p) = (2 aikViV^J ‚Ä¢ (2 ajiWjW^j - ^2 ajkvkw}j ‚Ä¢ aaviwl ^
= ^RijkiViwjvkwl, (2)
where
Rijkl Q-ik ' Qjl Hjk ' &il
are the 2 x 2 minors of II, corresponding to rows i, j and columns k, I. For example, #1234 = ¬´13 ‚Ä¢ ¬´24 - ¬´14 ‚Ä¢ ¬´23 comes from rows 1, 2 and columns 3, 4 of
11 =
flu fli2 0 13 014
a 12 fl22 023 ^24
R1212 = 011 ‚Ä¢ 022 ‚Äî 012 ‚Ä¢ 012 is the sectional curvature of the Xi,x2-plane.
R is called the Riemannian curvature tensor. Thus, the Rieman-nian curvature tensor is just the 2x2 minors of the second fundamental tensor.
Immediately,
Rjikl = Rijlk = Rijkl (4)
(interchanging two rows or columns changes the sign of the minor), and
Rklij ~ Rijkl
(5)
—Ç–ø-DIMENSIONAL SURFACES IN R" 35
because II is symmetric. One can easily check Bianchi‚Äôs first identity on permutation of the last three indices:
Ryu + Rikij + Rujk - 0. (6)
To obtain a definition of R independent of the choice of orthonormal coordinates on TPS, note that R is the bilinear form II –ª II on /\2TPS. Indeed, if {c,} gives a basis for TPS, so that {ek a e(: k< 1} gives a basis for /\2TPS, then
II –ª ll(ek A ei) = ll{ek) A 11(6,) = arke^j A as(e^j,
and
(<?, A ej) ‚Ä¢ II A ll(ek –ª ei) = aik ‚Ä¢ a,ji - a]k ‚Ä¢ au = Rijki.
As a bilinear form on /\2TPS, R is characterized by the values ¬£ ‚Ä¢ R(¬£) for unit 2-vectors ¬£ E /\2TPS. Actually R is determined by the sectional curvatures P ‚Ä¢ R(P) for 2-planes (unit simple 2-vectors).
The Ricci curvature Ric is a bilinear form on TPS, defined as a kind of trace of the Riemannian curvature. Just as the trace of a matrix [c,7] is a sum Sc,, over a repeated subscript, the coordinates Rjk of the Ricci curvature are given by
Rji = 2 Rijn- (7)
i
If you think of Rijki as a matrix of matrices,
[Rfi¬£i]
[RjmA:2] [R/mfem] _
then Rji is the corresponding matrix of traces, so the definition of Ric as a bilinear form does not really depend on the choice of orthonormal coordinates for TPS. Its application to ex yields the sum of the sectional curvatures of axis planes containing ex:
Ci ‚Ä¢ Ric (ci) = Rn = S Ran = ^ Run
i i* 1
m
= 2 K(e 1 A C,).
1 = 2
36 CHAPTER 5
Hence for any orthonormal basis ux,. .., vm for TPS,
m
vx ‚Ä¢ Ric (Ui) = 2 K(vx a Vi), (8)
i=2
and for any unit v G TPS,
v ‚Ä¢ Ric (v) = ‚Äî‚Äî~~~z f K(v a w). (9)
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