Gauss-Bonnet theorem Gaussian curvatures

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... 

Gauss-Bonnet Theorem. For an orientable dosed smooth surface Sin E, the total Gaussian curvature integrated over S is a topological invariant... 

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. 

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 have not yet discussed the gaussian curvature term K/R1R2 in Eq.l. A well-known theorem (Gauss-Bonnet) tells us that... 

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