Geometry, Topology, and Shape Representation - Mathematical Tools for Shape Analysis and Description - page 23

Let M be a real n -dimensional differentiable manifold in which the tangent space T p .M/
at any point p2M is equipped with an inner product (i.e., a symmetric positive definite bilinear
form ³ h;i p which varies smoothly from point to point. Once a local coordinate system is chosen,
every inner product h;i p is completely defined by inner products he i ;e j i p Dg i;j .p/ of elements
e 1 ;e 2 ;:::;e n of a standard basis in R n , that is, by the real symmetric and positive-definite nn
matrix .g i;j .p// , called a metric tensor . In fact, if xD.x 1 ;x 2 ;:::;x n / and yD.y 1 ;y 2 ;:::;y n /2
T p .M/ , we have

X

hx;yi p D g i;j x i y j :

i;j

e collection of these scalar products is called the Riemannian metric g with the metric tensor
.g i;j / . e length ds of the vector .dx 1 ;dx 2 ;:::;dx n / is expressed by the quadratic differential
form

X

ds 2 D g i;j dx i dx j

i;j

which is called the line element of the metric g .
A Riemannian manifold is a real n -dimensional differentiable manifold equipped with a
Riemannian metric. e length of a curve is expressed by

Z s X

l./D g i;j dx i dx j :

i;j

e intrinsic metric on a Riemannian manifold is defined as the infimum of the lengths of rectifi-
able curves joining two given points of the manifold. e Riemannian metric induces the intrinsic
metric of M n , so that the distance between two points p , q2M n is defined as

t X
i;j

Z Z

1 1

h d

dt ; d

g i;j dx i
dt

dx j
dt dt

i 1 2 D inf

inf

dt

0 0

where the infimum is taken over all rectifiable curves W0;1!M n connecting p and q (cf.
definition of geodesic distance and inner metric before).

³A bilinear form is a functionBWXX!Rwhich is linear in each argument separately. To every bilinear form it is possible
to associate a unique quadratic form that can be expressed in terms of a square matrix.