The Gauss-Bonnet theorem

Choose a triangle traced on a surface, whose three edges are geodesics. From the theorem, we have 

The external angles, 6i, are related to the internal angles, by = ic- Ui so that the area of the geodesic triangle is  

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The Euler characteristic is usually calculated in two ways by using the Gauss-Bonnet theorem [7,85,207,210,222] [Eq. (8)] or by combining the Cartesian and Gauss theorems [Eqs. (122) and (123)], which is also called the Euler formula [23,76,224]. 

The Euler characteristic, %, of a closed surface is related to the local Gaussian curvature K r) via the Gauss-Bonnet theorem [Eq. (8)]. A number of different schemes have been proposed to calculate the local curvatures and the integral in Eq. (8). 

To end this section, note that the Cartesian formula (120) is suitable to calculate the local Gaussian curvature while an exact value of the Euler characteristic is obtained from the Gauss-Bonnet theorem. 

The Gauss-Bonnet theorem, which relates integrals of Gaussian curvature (1/(/ii) in three dimensions) over a surface to integrals of mean curvature (1/iii + I/R2 in three dimensions) over boundaries of the surface, is particularly simple in two dimensions. In two dimensions, the (N — 6)-rule is equivalent to the Gauss-Bonnet theorem. 

There is a remarkable relationship between the average Gaussian curvature of a surface and its topology as quantified by the genus, which is the number of holes in a multiply connected surface. The relationship, the Gauss-Bonnet theorem, when applied to a surface of constant Gaussian curvature, is... 

Figure 1.8 Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (flj) and the geodesic curvature along the arcs AB, BC, CD and DA.

A more general, global version of the Gauss-Bonnet theorem can now be stated Let x be an oriented surface and R be a bounded region of x. As before, let the boimdary of R be the union of m simple curves that do not selfintersect, and let be the external angles at the m vertices. 

For instance, 3x = 0 <=> x a point of inflexion. Since 3a = 0 x G L/L, there will be 9 of these. Now via X C/L, we get flat metrics on X with curvature = 0. But if we instead look at a metric on X induced from the standard metric on P2 via X = C C P2, we get a metric whose curvature at the 9 points of inflexion equals that of P2, which is positive and by the Gauss-Bonnet theorem, it must be negative at other points. The wobbly curvature points up the fact that X does not fit symmetrically in P2 - we will discuss this further in Lecture III. Another indication of the antagonism between C/Z+Z-u and C is the Gelfand-Schneider result with a few exceptions for very special u s, (i.e., u G Q( /— n)), u and the coefficients of any isomorphic cubic C are never simultaneously algebraic. 

In fact, this result may be viewed to be the same as that of the Gauss-Bonnet theorem if we recall that the angle defects are essentially the net (i.e., integrated) Gaussian curvatures associated with each vertex. 

This in fact readily follows from the Gauss-Bonnet theorem, as indicated elsewhere. General results for the integrated absolute anisotropic curvature seem to be more difficult to obtain. But there is the ... 

The integral over the Gaussian curvature in Eq. [27] is a topological invariant.i "i85 For a closed orientable 2D surface (i.e., one without boundary), the Gauss—Bonnet theorem ties the value of this invariant to the genus g or the Euler—Poincare characteristic x of the surface ... 

We may now analyse the respective stabilities of the Lq and L3 structures in a qualitative way [5.4, 5.22]. Consider the horizontal and vertical axes of Fig. 5.22. K is relevant here because the and L3 structures have such contrasting topologies. We can estimate its contribution to the difference in elastic energies between the two phases by applying the Gauss-Bonnet theorem ... 

Remarkably, the topology is linked to the integral surface curvature by the following simple equation (the Gauss-Bonnet theorem) ... 

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