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where L
d
is the length of p
d
(t). If the curve corresponding to p(t) is closed and convex,
then
p
2
.
ALd
=
d
Proof.
See [FarN90a].
One can relate parallel curves and evolutes to the Gauss map. See [BaGM82].
9.8 Metric Properties of Surfaces
Now that we know a little about the metric properties of curves in
R
n
, we move on
to metric properties of Riemannian manifolds in general and surfaces in particular.
We need to generalize what we did in
R
n
, that is, our “local” view needs to be gener-
alized to a global one. Probably the most natural first use of the metric is to define
the length of curves in such manifolds and the notion of volume (area for surfaces).
We start with the length of a curve in a manifold.
Let g : [a,b] Æ
M
n
be a curve in an n-dimensional submanifold
M
n
of
R
k
. Recall
that, as a curve in
R
k
, the length of g, l
b
g, is just the integral
b
Ú
a
g¢
.
The only thing new will be how we interpret this formula. Note how the length
depends on the length of the tangent vectors of the curve. However, as a curve
in M
n
,
we do not want to think of the tangent vectors as vectors in
R
k
, but as vectors in the
tangent spaces of
M
n
that are invariants associated to a manifold. Therefore, the
length of a curve in a manifold should be thought of as a function of the dot product
in each tangent space for the manifold, so that it is determined by the Riemannian
metric for the manifold. Of course, as submanifolds of
R
k
we shall always be using
the Riemannian metric induced by the standard dot product on
R
k
. However, this par-
ticular choice is irrelevant and the reader should be aware of the fact that anything
we write down involving dot products would be valid for any other Riemannian
metric. Recall our comments at the beginning of the chapter that one should think of
manifolds in an intrinsic way and forget the engulfing space
R
k
.
Once we have a notion of length of a curve in a manifold, we can define the dis-
tance between its points.
Definition.
The
distance between two points
p
and
q
in a connected Riemannian
manifold
M
n
, denoted by dist(
p
,
q
), is defined by
{
}
(
)
=
b
gg
[
]
Æ
n
dist
pq
,
inf
l
:
a b
,
M
is a curve from to
p
q
.
It is not hard to show that the function dist(
p
,
q
) is a metric on the manifold
M
n
.
Since this metric was derived from a Riemannian metric, it is the latter that is funda-
mental to the metric properties of a manifold. In differential geometry the Riemannian
metric is usually brought into the picture using the following older terminology:
