Mathematical Principles of Theoretical Physics

3.1. BASIC CONCEPTS 111

Hence we have

̃gijd ̃xidx ̃j=(d ̃x^1 ,···,d ̃xn)( ̃gij)









dx ̃^1

..

.

dx ̃n







(3.1.10) 

=(dx^1 ,···,dxn)(aki)T(bki)T(gkl)(blj)(alj)







dx^1

..

.

dxn







=(dx^1 ,···,dxn)(gij)







dx^1

..

.

dxn







=gijdxidxj.

It follows that the metric (3.1.8) is invariant.

2. The length of a curve onMis determined by the metric{gij}. Letγ(t)be a curve
   connecting two pointsp,q∈M, andγ(t)be expressed by

(3.1.11) x(t) = (x^1 (t),···,xn(t)), 0 ≤t≤T, x( 0 ) =p,x(T) =q.

By (3.1.8) the infinitesimal arc-length is given by

ds=

√

gijdxidxj=

√

gij(x(t))x ̇ix ̇jdt.

Hence the lengthLofγ(t)is given by

(3.1.12) L=

∫q

p

ds=

∫T

0

√

gij(x(t))

dxi

dt

dxj

dt

dt.

The invariance ofdsas shown in (3.1.10) implies that the lengthLin (3.1.12) is independent
of the coordinate systems.

3. The volume of a bounded domainU⊂Mis invariant. The reason why{gij}is called
   a metric is that such quantities as the length, area, volume and angle are all determined by the
   metric{gij}.
   Given a Riemannian manifold{M,gij}, letU⊂Mbe a bounded domain. Then the
   volume ofUis written as

(3.1.13) V=

∫

U

Ω(x)dx.

where the volume elementΩdxreads

(3.1.14) Ω(x)dx=

√

−gdx, g=det(gij).

To derive (3.1.14), letM⊂Rn+^1 be an embedding, and

~r(x) ={r 1 (x),···,rn+ 1 (x)},