When does curvature create stretching in d dimensions?
eg. 3-dimensional sheet in 5 dimensions
dr
r(x1, x2, x3)
4
^
^
2
^
1
material
space
unstretched  ó g = 0
r1... r5
distance
ds2 = · dxi2
~
ds2 = dr ¥ dr
= ds2 + · dxi [ dr ¥ dr   - dij]  dxj 
dxi
dxj
~
strain tensor gij
two dxÕs curve in same direction
...stretched
But...
all dxÕs curve in different directions
...unstretched