# Riemannian geometry a beginners guide - Morgan F.

ISBN 0-86720-242-4

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n

H= â€” 1 Bni/dXi = â€”divn.

If the hypersurface is given as a level set {/(*i, ...,*â€ž) = 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 â€¢ 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 â€¢ â€¢ â€¢ 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 â€¢ â€¢ â€¢ (1 + | V|kw) dxm ~ (1 - V â€¢ H)dxx The formula follows.

dxr

5.2. Sectional and Riemannian curvature. The sectional curvature Ðš of S at p assigns to every 2-plane P Ð¡ TPS the Gauss curvature of the 2-dimensional surface

5ÐŸ(Ð 0Ð¢Ñ€5Ñ…).

If v, w give an orthonormal basis for P, then by its definition the sectional curvature is

K(P) = II(i>, u) â€¢ ll(w, w) - II(u, w) â€¢ II(d, w). (1)

For example, if II = [ai}] and P = ex Ð» e2 is the ^i,jr2-plane, then the sectional curvature is

K(P) = ll(eu ex) â€¢ II(e2, e2) - ll(eu e2) â€¢ II(eb e2)

= flu â€¢ a22 â€” aX2 â€¢ 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 â€¢ â€¢ â€¢ x Râ€ž-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 â€¢ (2 ajiWjW^j - ^2 ajkvkw}j â€¢ 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 = Â«13 â€¢ Â«24 - Â«14 â€¢ Â«23 comes from rows 1, 2 and columns 3, 4 of

11 =

flu fli2 0 13 014

a 12 fl22 023 ^24

R1212 = 011 â€¢ 022 â€” 012 â€¢ 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)

Ñ‚Ð¿-DIMENSIONAL SURFACES IN R" 35

because II is symmetric. One can easily check Bianchiâ€™s 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 Ð» 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 Ð» ll(ek A ei) = ll{ek) A 11(6,) = arke^j A as(e^j,

and

(<?, A ej) â€¢ II A ll(ek Ð» ei) = aik â€¢ a,ji - a]k â€¢ au = Rijki.

As a bilinear form on /\2TPS, R is characterized by the values Â£ â€¢ R(Â£) for unit 2-vectors Â£ E /\2TPS. Actually R is determined by the sectional curvatures P â€¢ 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,

[RfiÂ£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 â€¢ 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 â€¢ Ric (Ui) = 2 K(vx a Vi), (8)

i=2

and for any unit v G TPS,

v â€¢ Ric (v) = â€”â€”~~~z f K(v a w). (9)

vol Sm J

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