Graphics Reference
In-Depth Information
Definition.
The quadratic form Q
I
defined on the tangent space T
p
(
M
n
) of the
manifold
M
n
with Riemannian metric • by
()
=∑
Q
I
vvv
is called the
first fundamental form
of the manifold at
p
.
We conclude the following:
9.8.1. Theorem.
The metric properties of a manifold are completely determined by
its first fundamental form.
Next, we want to bring parameterizations into our discussion, because this is how
manifolds are defined intrinsically, and we do not want to rely on them being con-
tained in some
R
k
. Locally, a manifold
M
n
looks like (can be identified with)
R
n
. For-
mally, such an identification corresponds to a parameterization. The question is, if we
do our work in
R
n
, how can one use the parameterization to translate the results back
to
M
n
? We begin by answering this question for lengths of curves.
Let
U
be an open set in
R
n
and let F(u
1
,u
2
,...,u
n
),
ÆÕ
n
,
F :
UVM
be a parameterization for a neighborhood
V
of a point
p
on the manifold
M
n
in
R
k
.
Assume that
U
contains the origin and that F(
0
) =
p
. It is easy to see that every curve
g(t) in
V
through
p
can be expressed in the form
()
=
(
()
)
g
t
F
m
t
,
where
(
)
Æ
n
()
=
m
:
-
a a
,
R
and
m
0
0
.
See Figure 9.16. How does the length of m relate to the length of g?
M
p
V
F
g
v
U
-a
0
a
u
t
Figure 9.16.
Representing curves in
surfaces.
m
