San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

The Intersection of Manifolds Satisfying
the Gauss-Bonnet Theorem and the
Surfaces Thereby Created

Here is a statement of the Gauss-Bonnet Theorem.

Let M be a compact two-dimensional Riemannian manifold with boundary ∂M. Let K(p) be the Gaussian curvature of M at point p, and let kg and the Surfaces Thereby Created be the geodesic curvature of ∂M. Then

∫∫M K(p)dA + ∫∂Mkgds = 2πχ(M)

where dA is the element of area of the surface, ds is the line element along the boundary of M, and χ(M) is the Eulers-Poincaré characteristic of M.

There is the matter of the sign of the integral of geodesic curvature that needs to be considered and this is best elucidated with the relatively easy topic of spheres.

The Gauss-Bonnet Theorem
for Portions of Sphere Created by Truncation

What is considered here are surfaces shaped like the surface of a lens and having a sharp edge where the two spherical surfaces meet. Such a lens-shaped surface can be formed by the intersection of two spheres. Let R1 and R2 be the radii of the two spheres and hence also the radii of curvature of the of the two lens surfaces. In the geometric scheme used here a portion of a sphere is the surface between a latitudinal angle of −π/2. Let Θ1 and Θ2 be the latitudinal angles for the lens edge where the two spheres intersect. Both Θ1 and Θ2 will be negative for the usual lens shape.

The radius r of the circle of intersection is given by

r = R1cos(Θ1) = R1cos(Θ1)

This is the radius of the sharp edge of the lens-like surface.

Crucial variables for the analysis are the cutoff distances

a1 = R1sin(Θ1)
a2 = R2sin(Θ2)

For the usual lens shape a1 and a2 are negative.

The Gaussian curvatures on the two lens faces are 1/R1² and 1/R2². Therefore the integrals of Gaussian curvature over the lens faces are just the surface areas times these curvatures.

The areas A1 and A2 of the lens surfaces are given by

A11) = 2πR1²(1 + sin(Θ1))
and
A22) = 2πR2²(1 + sin(Θ1))

and thus the integrals of Gaussian curvature on the lens faces are

A11)/R1² = 2π(1 + sin(Θ1))
and
A22)/R2² = 2π(1 + sin(Θ1))

Geodesic Curvature on
the Circle of Intersection

The geodesic curvature of a circle of radius r on a sphere of radius R is

kg = −a/(rR)

Therefore the integral of geodesic curvature around the circumference of 2πr is

−2πa/R
and since a=Rsin(Θ)
this reduces to
−2πsin(Θ)

The sum of the integral of Gaussian curvature over the lens face of radius of curvature R1 and the integral of geodesic curvature of the circle of intersection within the sphere of radius R1 is then

S1K(p)dA + ∫S1kg1ds
= 2π(1 + sin(Θ1)) − sin(Θ1) = 2π

and likewise for the second sphere; i.e.,

S2K(p)dA + ∫S2kg2ds
= 2π(1 + sin(Θ2)) − 2πsin(Θ2) = 2π

The lens surface is topologically equivalent to a sphere so its Euler characteristic χ is equal to 2. Therefore the sum of these two terms is 4π which is just 2πχ(lens). Thus the integral of Guassian curvature over the surface of the lens plus the two integrals of geodesic curvature around the edge with the geodesic curvature determined with respect to the two surfaces which contain it is equal to 2π times the Euler characteristic of the lens.

In the tabulation for the lens the parameters a1 and a2 were negative so the terms −2πsin(Θ1) and −2πsin(Θ2) are positive and add to the quantities on the LHS of the equation. If a1 and a2 are negative then the surface has an indented crease instead of a protruding edge and the contribution of the integrals of geodesic curvature are negative.

The More General Case of the Surfaces
Created by the Intersection of Manifolds
Satisfying the Gauss-Bonnet Theorem

Let M1 and M2 be two closed manifolds satisfying the conditions for the Gauss-Bonnet Theorem. Let M1 and M2 be situated such that they intersect in the curve C. The curve C then divides M1 and M2 into manifolds with boundaries and for each the boundary is C. All of them are topologically equivalent to a disk and therefore their Euler characteristics are all 1. Let the manifolds be denoted as M1a, M1b, M2a and M2b. Each of these satisfy the the Gauss-Bonnet Theorem. Therefore for each

∫∫MiKdA + ∫C,Mikgds = 2π

The geodesic curvature depends not only on the curve C but the surface it is located in.

There are four surfaces thereby created; i.e. M1a∪M2a, M1a∪M2b, M1b∪M2a, M1b∪M2b. These surfaces are not necessarily smooth; they may have a sharp edge or indentation at C and therefore do not satisfy the standard Gauss-Bonnet Theorem.

For each combination Mi∪Mj the Euler characteristic is 2. Furthermore

∫∫MiKdA + ∫C,Mikgds + ∫∫MjKdA + ∫C,Mjkgds = 2π + 2π = 4π

This reduces to

∫∫Mi∪MjKdA + ∫C,Mikgds + ∫C,Mjkgds = 4π = 2πχ(Mi∪Mj)

Therefore the Gauss-Bonnet Theorem holds for a surface with a sharp edge or indentation provided the integral of geodesic curvature of the edge or indentation is evaluated with respect to both surfaces it is contained in. It is uncertain whether the same holds true for manifolds whose characteristics are different from 2.

For more on aspects of the Gauss-Bonnet Theorem.

HOME PAGE OF applet-magic