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The Intersection of Manifolds Satisfying the Gauss-Bonnet Theorem and the Surfaces Thereby Created |
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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 k_{g} and the Surfaces Thereby Created be the geodesic curvature of ∂M. Then
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.
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 R_{1} and R_{2} 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 Θ_{1} and Θ_{2} be the latitudinal angles for the lens edge where the two spheres intersect. Both Θ_{1} and Θ_{2} will be negative for the usual lens shape.
The radius r of the circle of intersection is given by
This is the radius of the sharp edge of the lens-like surface.
Crucial variables for the analysis are the cutoff distances
For the usual lens shape a_{1} and a_{2} are negative.
The Gaussian curvatures on the two lens faces are 1/R_{1}² and 1/R_{2}². Therefore the integrals of Gaussian curvature over the lens faces are just the surface areas times these curvatures.
The areas A_{1} and A_{2} of the lens surfaces are given by
and thus the integrals of Gaussian curvature on the lens faces are
The geodesic curvature of a circle of radius r on a sphere of radius R is
Therefore the integral of geodesic curvature around the circumference of 2πr is
The sum of the integral of Gaussian curvature over the lens face of radius of curvature R_{1} and the integral of geodesic curvature of the circle of intersection within the sphere of radius R_{1} is then
and likewise for the second sphere; i.e.,
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 a_{1} and a_{2} were negative so the terms −2πsin(Θ_{1}) and −2πsin(Θ_{2}) are positive and add to the quantities on the LHS of the equation. If a_{1} and a_{2} 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.
Let M_{1} and M_{2} be two closed manifolds satisfying the conditions for the Gauss-Bonnet Theorem. Let M_{1} and M_{2} be situated such that they intersect in the curve C. The curve C then divides M_{1} and M_{2} 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 M_{1a}, M_{1b}, M_{2a} and M_{2b}. Each of these satisfy the the Gauss-Bonnet Theorem. Therefore for each
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. M_{1a}∪M_{2a}, M_{1a}∪M_{2b}, M_{1b}∪M_{2a}, M_{1b}∪M_{2b}. 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 M_{i}∪M_{j} the Euler characteristic is 2. Furthermore
This reduces to
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.
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