Page 385 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 30.6.2 Interpretation and Significance

The theorem applies in particular to compact surfaces without boundary, in which case the
integral

M  k ds
g

can be omitted. It states that the total Gaussian curvature of such a closed surface is equal to 2
times the Euler characteristic of the surface. Note that for orientable compact surfaces without
boundary, the Euler characteristic equals 2 – 2g, where g is the genus of the surface: Any orientable
compact surface without boundary is topologically equivalent to a sphere with some handles
attached, and g counts the number of handles.
If one bends and deforms the surface M, its Euler characteristic, being a topological invariant,
will not change, while the curvatures at some points will. The theorem states, somewhat
surprisingly, that the total integral of all curvatures will remain the same, no matter how the
deforming is done. So for instance if you have a sphere with a “dent”, then its total curvature is
4 (the Euler characteristic of a sphere being 2), no matter how big or deep the dent.
Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a
non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic
1: the Gauss–Bonnet formula does not work. It holds true, however, for the compact closed unit
disc, which also has Euler characteristic 1, because of the added boundary integral with value 2.

As an application, a torus has Euler characteristic 0, so its total curvature must also be zero. If the
torus carries the ordinary Riemannian metric from its embedding in R , then the inside has
3
negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature
is indeed 0. It is also possible to construct a torus by identifying opposite sides of a square, in
which case the Riemannian metric on the torus is flat and has constant curvature 0, again resulting
in total curvature 0. It is not possible to specify a Riemannian metric on the torus with everywhere
positive or everywhere negative Gaussian curvature.
The theorem also has interesting consequences for triangles. Suppose M is some 2-dimensional
Riemannian manifold (not necessarily compact), and we specify a “triangle” on M formed by
three geodesics. Then we can apply Gauss–Bonnet to the surface T formed by the inside of that
triangle and the piecewise boundary given by the triangle itself. The geodesic curvature of
geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the
sum of the turning angles of the geodesic triangle is equal to 2 minus the total curvature within
the triangle. Since the turning angle at a corner is equal to ð minus the interior angle, we can
rephrase this as follows:

The sum of interior angles of a geodesic triangle is equal to ð plus the total curvature enclosed
by the triangle.
In the case of the plane (where the Gaussian curvature is 0 and geodesics are straight lines), we
recover the familiar formula for the sum of angles in an ordinary triangle. On the standard
sphere, where the curvature is everywhere 1, we see that the angle sum of geodesic triangles is
always bigger than ð.

30.6.3 Special Cases

A number of earlier results in spherical geometry and hyperbolic geometry over the preceding
centuries were subsumed as special cases of Gauss–Bonnet.

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