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normal vector, one cannot talk about any Gauss map nor define the Gauss curvature
directly as before, but now one would be able to prove that the second structural equa-
tion holds for some function K and this function would then be defined to be the
Gauss curvature
. Now, the forms w
ij
and q
i
depend on the frame field, but the func-
tion K turns out to be independent of it. In the end, one would therefore again have
the first and second structural equations hold. Hence one would get the same geo-
metric consequences as before. The Christoffel symbols are also embedded in all this.
See [Spiv70b].
9.17
Where to Next?
In this chapter we have taken the first steps in learning about some of the beautiful
geometric results as seen through the eyes of a differential geometer. From a histori-
cal perspective, our presentation may have been more modern but we did not learn
much more than what was already known at the time of Gauss. We have basically
looked at results that can be deduced from the first and second fundamental forms.
We have seen how a “simple” map like the Gauss map can contain within it a wealth
of information.
Euler's formula about the Euler characteristic showed that some properties of
spaces did not depend on their metric. Gauss showed that Gauss curvature depended
only on arc-length in the surface. Another important early result is the Gauss-Bonnet
theorem below, which generalizes the theorem in the plane that the sum of angles of
a triangle is p. It shows a connection between the metric invariant, Gauss curvature,
and a topological invariant, the Euler characteristic, and generalizes the relationship
between the total curvature of a closed curve and its degree (Theorem 9.3.13 and
Corollary 9.3.14).
9.17.1. Theorem.
(Gauss-Bonnet) Let
S
be a compact, closed, orientable surface.
Then
ÚÚ
()
K
=
2pc
S
,
(9.95)
S
where K is the Gauss curvature function on
S
and c(
S
) is the Euler characteristic of
S
. (The integral in this equation is called the
total Gauss curvature
of
S
.)
Proof.
See one of the references for differential geometry.
One of the interesting consequences of this theorem is what it says about the
Gauss curvature function. Since the right-hand side of Equation (9.95) is a topologi-
cal invariant, no matter how we deform a surface, there are constraints as to how the
Gauss curvature distributes itself. It is not a totally arbitrary function. Here is an appli-
cation of the Gauss-Bonnet theorem.
9.17.2. Corollary.
A compact, closed, orientable surface whose Gauss curvature is
positive everywhere must be homeomorphic to a sphere.
