Page 386 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

1. Triangles Notes

In spherical trigonometry and hyperbolic trigonometry, the area of a triangle is proportional to
the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse)
amount by which its exterior angles fail to add up to 360°.
The area of a spherical triangle is proportional to its excess, by Girard’s theorem – the amount by
which its interior angles add up to more than 180°, which is equal to the amount by which its
exterior angles add up to less than 360°.

The area of a hyperbolic triangle conversely is proportional to its defect, as established by
Johann Heinrich Lambert.

2. Polyhedra

Descartes’ theorem on total angular defect of a polyhedron is the polyhedral analog: it states
that the sum of the defect at all the vertices of a polyhedron which is homeomorphic to the
sphere is 4. More generally, if the polyhedron has Euler characteristic  = 2 – 2g (where g is the
genus, meaning “number of holes”), then the sum of the defect is 2. This is the special case of
Gauss–Bonnet, where the curvature is concentrated at discrete points (the vertices).
Thinking of curvature as a measure, rather than as a function, Descartes’ theorem is Gauss–
Bonnet where the curvature is a discrete measure, and Gauss–Bonnet for measures generalizes
both Gauss–Bonnet for smooth manifolds and Descartes’ theorem.

30.6.4 Combinatorial Analog

There are several combinatorial analogs of the Gauss–Bonnet theorem. We state the following
one. Let M be a finite 2-dimensional pseudo-manifold. Let (v) denote the number of triangles
containing the vertex v. Then

 v int M (6   (x))   v M (4   (v)) 6 (M),



where the first sum ranges over the vertices in the interior of M, the second sum is over the
boundary vertices, and (M) is the Euler characteristic of M.
More specifically, if M is a closed digital 2-dimensional manifold, The genus
g = 1 + (M + 2M – M )/8,
6
3
5
where M indicates the set of surface-points each of which has i adjacent points on the surface. See
i
digital topology
30.6.5 Generalizations

Generalizations of the Gauss–Bonnet theorem to n-dimensional Riemannian manifolds were
found in the 1940s, by Allendoerfer, Weil and Chern–Weil homomorphism. The Riemann–Roch
theorem can also be considered as a generalization of Gauss–Bonnet.
An extremely far-reaching generalization of all the above-mentioned theorems is the Atiyah–
Singer index theorem.

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