When does curvature create stretching in d dimensions?
eg. 3-dimensional sheet in 5 dimensions
d
r
r
(x
1
, x
2
, x
3
)
4
^
^
2
^
1
material
space
unstretched
ó
g
= 0
r
1
... r
5
distance
ds
2
=
·
dx
i
2
~
ds
2
= d
r
¥
d
r
= ds
2
+
·
dx
i
[
d
r
¥
d
r
-
d
ij
]
dx
j
dx
i
dx
j
~
strain tensor
g
ij
two dx
Õ
s curve in
same
direction
...
stretched
But...
all dx
Õ
s curve in
different
directions
...un
stretched
