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Theorem 9.9.27 is the reason that mapmakers had so much trouble making a good
map of the world. The converse of Theorem 9.9.27 is false in general, namely, a Gauss
curvature-preserving map between surfaces is not necessarily an isometry. The
converse
is
true in one special case, namely, the constant curvature case.
9.9.28. Theorem.
Any two surfaces having the same constant Gauss curvature are
isometric.
Proof.
See [Stok69].
We finally come to our main result, namely, that knowledge of the first and second
fundamental forms of a surface determine it completely locally.
9.9.29. Theorem.
(The Fundamental Theorem of Surfaces) Let E, F, G, L, M, and
N be differentiable functions defined on an open set
U
in
R
2
with E, G > 0. Assume
(1) EG - F
2
> 0 and
(2) E, F, G, L, M, and N satisfy the compatibility Equations (9.61) and (9.62).
Then, for each
p
in
U
there exists an open neighborhood
V
of
p
in
U
, a surface
S
in
R
3
, and a diffeomorphism
F :
VS
Æ
,
so that E, F, G, and L, M, N are the coefficients of the first and second fundamental
form of
S
, respectively. Furthermore, the surface
S
is unique up to rigid motion.
Proof.
See [DoCa76], [Lips69], or [Spiv75]. The functions E, F, and G need only be
C
2
and the functions L, M, and N, only C
1
.
9.10
Geodesics
This section takes a look at how we might generalize the concept of a “straight” line
that we have in
R
n
. The generalization will be called a geodesic. We shall discuss geo-
desics in a surface in
R
3
. The reason for concentrating on this special case is that it
allows us to introduce the subject of geodesics in a geometrically intuitive way. The
generalization to higher-dimensional manifolds with a Riemannian metric involves
more advanced concepts from differential geometry that we shall only briefly get to
in Section 9.17. We shall present four definitions for a geodesic. Although geodesics
depend only on intrinsic properties of a surface, not all the definitions will have that
property and will seem to rely on a surface's imbedding in
R
3
. In practice this will not
be a problem. In fact, it will facilitate certain computations.
Since a manifold is not a vector space, generalizing the simple definition of a
straight line in terms of a point and direction vector will clearly not work. To gener-
alize, one must understand what one is generalizing. What are some intuitive prop-
erties that one usually associates to a straight line? Here are several:
