# Riemannian geometry a beginners guide - Morgan F.

ISBN 0-86720-242-4

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To check invariance, we must show that if u\ u1' give coordinates at p, then

P = A{AtA)~xAt = Ag~1At.

^dxn a dxm i,i du1 ^ dul â€™

to obtain

ildxm d2xm du1 ^ du! duÃ® duk

Therefore

X[, = Xij + lrikXt,

k

where

r;* = 2 gilxjk â€¢ x/

I

= 2 gl7d)[(x, â€¢ Xj)k + (x, â€¢ Xk)j - (Xj â€¢ X^)/]

I

- 2^ gll{glj,k + glk,j ~ gjkJ)â€¢

INTRINSIC RIEMANNIAN GEOMETRY 45

where we henceforth agree to sum over repeated indices. (Getting such formulas rightâ€”knowing whether the du" goes on the top or the bottomâ€”is perhaps the hardest part of linear algebra, but our index conventions make it automatic.) This verification is something of a mess, but here we go. First we note that

and similar equations hold for -gjkj, gkijâ€¢ Combining all three with the definition of ryk and

Hence

gjl.k

d2Ur dus dur d2Us

lj' duk> du1' duj' du1' duk'_

â€” â€” gpq

dup duq

yields

[grs,t ~ grt,s + gte.r] + H I ,

where

^ grs

d2Ur dus duâ€™â€™ dukf du1'

dur d2Us d2Ur dus

duif du1' duk' du}' du1' duk' ur dus dur d2us

duJ' du1' duk' duJ' du1'_

dur d2Us d2Ur

duj' duk' du1' duk' dl f d2Ur dus

duk' duJf du1'

because grs is symmetric.

46 CHAPTER 6

Therefore

Ñ‚,.-, Ð†Ð´Ð¸1' dur du* .

Ñ– dV duh duj' duk' bup 4

where 5^ = 1 if s = q and 0 otherwise. Since Sqgpqgrs - gpqgqr -

d2Ur du1'

_ du1' dur du1 .

'* au? du>'auk'\2g sn,. + g.,r

du" dur du' p d2uh du1'

1 rt "f"

+

duâ€™â€™ duk' dur

dup du1' Ð´Ð¸Ðº' ÐŸ duâ€™â€™ duk' duh â€™

by changing the dummy index in the last term from r to h.

Multiplying both sides by duh/dutf and changing primes and indices yields

d2U1' _ h du1' _ j-,t- , duh' duk' dum du1 lm duh Hk du1 dum

(2)

Now

Ñ– t \rkt

xy-TT,x' +r ykx

duJ

d (^xm^ + ryk(^â€”x

du1â€™ \du Xmn

- <>Un dU1â€™ | x,

du1' dur

durn d2u1' du1 dum du1 du1'

+ xr

du

kt

dur

Ð“''

1 jk-

By (2),

X\'j = Xmn

dun du1

du1' dur

+ XK

' h du1' du1 ,

lm u i. *â– hk

du du1'

duh' duk' du1 du1 dum du1'

+ Ð“ yk

du

kt

du'

Ñ–^Ð´Ð¸Ð£ h bumbui' ,n d^du1' lm

By changing dummy variables in the second term (mâ€”>k, h -*m, I â€”>n), we obtain

INTRINSIC RIEMANNIAN GEOMETRY 47

t __ dti \Xm + Ym X^\ â€” xm

J dum dllJ' nk dum duj'

as desired.

Remark. Of the two proofs, the first has the advantages of being shorter and deriving the formula, whereas the second proves a given formula.

6.3. Geodesics. Let C be a C2 curve in a C2 m-dimensional surface S in R", with curvature vector k at a point p E C. We define the geodesic curvature Kg as the projection of k onto the tangent space TPS. Equivalently, Kg is the covariant derivative of the unit tangent vector. While curvature k is extrinsic, geodesic curvature Kg is intrinsic.

A geodesic is a curve with = 0 at all points. For example, geodesics on spheres are arcs of great circles, but other circles of latitude are not geodesics. (See Figure 6.2.) Shortest paths turn out to be geodesics, but there are sometimes also other longer geodesics between pairs of points. For example, nonantipodal points on the equator are joined by a short and a long geodesic, depending on

Figure 6.2. On the sphere, great circles are geodesics (Kg = 0), but other circles of latitude are not (Kg Â¥= 0).

48 CHAPTER 6

which way you go. The poles are joined by infinitely many semicircular meridians of longitude, all of the same length.

The following theorem explains why shortest paths must be geodesics.

6.4. Theorem. A curve is a geodesic if and only if the first variation of its length vanishes.

Proof. Let x(t) be a local parametrization by arc length. Corresponding to an infinitesimal, compactly supported change 8x in x(t) is a variation in length

8L = 8 J (x â€¢ x)m = Ji(x â€¢ x)â€œ1/22x â€¢ 8x

= Jt-8x=-Jt-8x= â€” J k â– 8x = â€” J Kg â– 8x

by integration by parts and the fact that 8x stays in the surface. 8L vanishes for all 8x along the surface if and only if Kg = 0 and the curve is a geodesic.

Remark. It follows from the theory of differential equations that in any C2 m-dimensional surface, through any point in any direction there is locally a unique C2 geodesic. Such a geodesic provides the shortest path to nearby points. A little more argument shows that if S is connected and compact (or connected and merely complete), between any two points some geodesic provides the shortest path (the Hopf-Rinow Theorem, see [CE, ch. 3] or [He, Theorem 10.4]).

6.5. Formula for geodesics. In local coordinates ul,... , um, consider a curve u(t) parametrized by arc length so that the unit tangent

vector T = u. The derivative of any function f(u) along the curve is

given by 'LfjUi (the chain rule). The covariant derivative of any vectorfield X1 along the curve satisfies

2 X\juj = 2 X[jUJ + 2 Y)kuâ€™Xk (1)

j } M

= xi + 2 r }ku'xk

M

INTRINSIC RIEMANNIAN GEOMETRY 49

[see 6.0(1)]. Hence for a geodesic (parametrized by arc length), the covariant derivative along the curve of the vectorfield X1 = T' = ul must vanish:

0 = ul + 1riJkuiuk. (2)

M

6.6. Hyperbolic geometry. As an example in Riemannian geometry, we consider 2-dimensional hyperbolic space H for which global coordinates are given by the upper halfplane

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