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The Gaussian Curvature and the
Angular Deficit of a Point of a Surface

The Gauss-Bonnet Theorem is a marvelous result that says that the integral of the Gaussian curvature over a closed smooth surface is equal to 2π times Ξ, the Euler characteristic of the surface. For any surface topologically equivalent to a sphere the Euler characteristic is equal to 2, Therefore the integral of the Gaussian curvature over such a surface is equal to 4π

There is an equally marvelous extension to the Gauss-Bonnet Theorem for a surface with a finite number of points of singularity where the curvature is infinite. According to that extension

SKdS + ΣCαj = 2πΞ

where αj is the angular deficit of the j-th point of singularity and C denotes the set of such points. The angular deficit of a point is the amount by which the sum of the angles of the surfaces impinging upon the point falls short of 2π radians. It can be visualized as a cutting out (excision) and unrolling or unfolding of the surface within a fixed distance from the point.

A special case of this theorem is its application to polyhedra. The Gaussian curvature on the faces of the polyhedron is zero. The angular deficit at at a vertex just 2π less the sum of the interior angles of the faces meeting at that vertex. Thus the sum of the angular deficits of the vertices of a regular polyhedron is thus equal to 4π. because the Euler characteristic of a regular polyhedron is 2. This property of polyhedra was discovered by René Descartes in the 17th century.

It would be nice aesthetically to be able to specify a quantity for the points of surface such that for all the points where the surface is smooth and differentiable it is the Gaussian curvature and at all points of singularity it is the angular deficit. Before analysis it might seem possible that even at the points where the Gaussian curvature is defined that it is equal an appropriately defined angular deficit. The analysis below will show that this is not true.

Consider a sphere of radius R. At each point the Gaussian curvature is 1/R². At any point draw a circle of radius r where r is measured within the spherical surface. Excise the disk circumvended by that circle, cut it along a straight line from the point to the circular edge and flatten it. The angle θ, in radians, subtended by the radius r is (r/R). The circumference of the circle is

2πRsin(θ)

The flattened excised disk is enclosed within a circle of radius r whose circumference is 2πr. The angular deficit α is then

α = [2πr − 2πRsin(θ)]/r = 2π[1 − (R/r)sin(θ)]
= 2π[1 − sin(θ)/(r/R)] = 2π[ 1 − sin(θ)/θ]

The function sin(θ)/θ is usually denoted as sinc(θ). The graph of α(θ) is

The series representation of sin(θ) is

sin(θ) = θ − θ3/3! + θ5/5! − θ7/7! + …
and hence
sinc(θ) = 1 − θ2/3! + θ4/5! − θ6/7! + …
and thus
α = 2π[θ2/3! − θ4/5! + θ6/7! + …]

Since θ is equal to r/R this means that

α ≅ (2πr²/3!)(1/R2)

Thus at any point for r≠0 there is an angular deficit that is approximately proportional to the Gaussian curvature. But the constant of proportionality depends upon r and goes to zero as r→0. Thus the Gaussian curvature cannot be depicted as something in the nature of an angular deficit.

What would work would be a ratio of the angular deficit per unit area of the disk excised; i.e.,

α/(πr²) = (1/3)(1/R2)

The points along an edge were purposely neglected in the above because they do not affect the results. At any point on a straight edge if a circle of radius r is excised the result is a disk with a fold. When the disk is unfolded it is exactly a circular disk and there is no angular deficit. Thus the angular deficit at the points at all points of an edge except the end points is zero.

So the extension of the Gauss-Bonnet Theorem requires the separate treatment of the points where the surface is smooth and differentiable and at the points of singularity of the curvature.

The Analogous Situation for Curves with Corners

Consider the movement around a curve. For the smooth portions of the curve the angle between the tangent to the curve and the horizontal can to called the turning angle. At the corners of the curve the turning angle goes through discontinuous jumps. The traversing of the circuit of the curve results in the turning angle going though an increase of exactly 2π radians or 360°.

The increase in the turning angle can be represented as a Stieltges integral of the turning angle function. Where the turning angle function is differentiable the integrand is the derivative of the turning angle function. Where it is not differentiable the integrand is the turning angle at the corner point. The Stieltjes integral provides a neat representation that incorporates the smooth and discontinuous points into one symbol.

Conclusion

The extension of the Gauss-Bonnet Theorem to include points of singularity of the curvature requires the separate treatment of the points where the surface is differentiable and the Gaussian curvature is defined and the points of singularity of the curvature. The two types of points may be included in a Stieltjes-type integral.

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