9.4: The Euler Characteristic
If we have any triangulation of a surface (i.e. a division of the surface into triangles), the Euler Characteristic is defined as the number of vertices minus the number of edges plus the number of faces of the polygons in the triangulation.
χ(M2) = #V - #E + #F
There is a corollary of the Gauss-Bonnet theorem, that we just developed in the last section, which states that the total curvature of a closed surface is equal to 2π times its Euler Characteristic.
M2KdA = 2πχ(M2)
The proof of this corollary can be seen here.
Demonstration
In this demonstration we draw a closed surface X(u,v) in one window and the triangulation of this surface in a second window. The number of divisions in the u- and v-directions can be changed by typing in different values for Steps1 and Steps2. In the control panel, we count the number of vertices, edges, and triangles (faces) in the triangulation. The Euler characteristic of the surface is calculated from these numbers.
Exercises
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1. What is the Euler Characteristic of an ellipsoid? How about a torus?
2. A torus is topologically equivalent to a sphere with one handle. We can also talk about closed surfaces that are topologically equivalent to spheres with multiple handles. Let g(M2) denote the number number of handles of an embedded surface. Express the total Gaussian curvature of a sphere with g-handles in terms of g.