11 The Theorema Egregium and the Gauss–Bonnet Theorem

11 The Theorema Egregium and the Gauss–Bonnet
Theorem
Definition 11.1 A property is called intrinsic if it preserved by local isometries, otherwise
it is called extrinsic.
Theorem 11.2 (Theorema Egregium of Gauss) The Gauss curvature is intrinsic.
Distance is not intrinsic in this sense as it is not always preserved by local isometries,
however
Theorem 11.3 Distance is preserved by (global) isometries, i.e. if ϕ : M → M ′ is a
(global) isometry, then
d(p, q) = d(ϕ(p), ϕ(q)) (p, q ∈ M) .
Definition 11.4 Let M be a surface in R 3 . Let X : U → W = X(U), (u, v) ↦→
X(u, v), be a local parametrization of a closed subset W of M. The area of W is
defined by
∫∫
A(W ) =
U
|X u × X v | du dv .
Lemma 11.5 The area of W is independent of the parametrization chosen.
Write dA = |X u × X v | du dv; this is called the element of (surface) area (another
common notation is dS). Then we can write
∫∫
A(W ) =
W
dA .
More generally, for any smooth function f : W → R we define
∫∫ ∫∫
f dA = ˆf(u, v) |X u × X v | du dv
W
U
where ˆf = f ◦ X is the coordinate expression of f. This is called the surface integral
(over W ) of f or the integral of f with respect to area; it is also independent of the
parametrization.
For integrals over regions W not covered by a single chart (=local parametrization),
if we can chop W into disjoint pieces W i each covered by single chart, then we add up
the results, i.e,
A(W ) = ∑ ∫∫
A(W i ) , f dA = ∑ ∫∫
f dA .
i
W
i
W i
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The sums can be shown to be independent of how we chop up W . We can integrate
over the whole surface provided it is compact in the following sense:
Definition 11.6 A surface M in R 3 is called compact (or, confusingly, closed) if
(i) it is a closed subset of R 3 in the sense of topology, i.e., its complement is an open
subset of R 3 , and (ii) it is bounded, i.e., contained in some ball of R 3 .
In particular we can define
Definition 11.7 The total curvature of M is the integral of its Gauss curvature
with respect to area:
∫
M
K dA .
We now define a topological invariant of a compact surface.
Theorem 11.8 Any compact surface can be triangulated.
Definition 11.9 The Euler characteristic of a compact surface M is
χ(M) = e(M) = F − E + V
where F = the number of faces, E = the number of edges, and V = the number of
vertices of some triangulation of M.
Theorem 11.10 The Euler characteristic of a compact surface M does not depend on
the triangulation chosen.
Theorem 11.11 If M and M ′ are homeomorphic compact surfaces, then they have
the same Euler characteristic.
Theorem 11.12 Any compact orientable surface M is homeomorphic to a sphere with
g handles for some g ∈ {0, 1, 2, ...}.
The number g is called the genus of M.
Theorem 11.13 The Euler characteristic of a compact orientable surface of genus g
is 2 − 2g.
Theorem 11.14 (Gauss–Bonnet) The total curvature of a compact surface is 2π
times its Euler characteristic, i.e.,
∫
M
K dA = 2πχ(M) .
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