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 be the geodesic curvature of ∂M. Then

∫_M K(p)dA + ∫_∂Mk_gds = 2πχ(M)

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.

The matter of the direction of traversal of the ∂M needs to be specified. That direction of traversal is counterclockwise when M is on the left.

The Gauss-Bonnet Theorem
for a Truncated Sphere

The Gaussian curvature of a sphere of radius R is 1/R at each point of its surface. Therefore a sphere of unit radius has Gaussian curvature equal to unity at all points. Consider the unit radius sphere given by

x² + y² + z² =1
and its intersection with the plane
z = a
where −1<z<1

The intersection is the circle given by

x² + y² = 1 − a²

The Geodesic Curvature of the Bounding
Circle of a Truncated Sphere

The geodesic curvature k_g on this circle in the unit sphere is given by

k_g = −a/(1−a²)^½

The integral of geodesic curvature around the circle of intersection is then

∫_C k_gds = [ −a/(1−a²)^½][2π(1−a²)^½]
= −2πa

In a sense the geodesic curvature represents the deviation of a curve from a geodesic of the surface. A geodesic on a sphere is a great circle, a circle whose center of curvature is the same as the center of the sphere. If a=0 then the circle of intersection is an equatorial circle, a geodesic, and, according to the above formula, the integral of the geodesic curvature is equal to zero.

The Surface Integral of Gaussian
Curvature on a Truncated Sphere

The integral of Gaussian curvature on the section of the sphere between z=−1 and a is, for the unit sphere, just equal to the area of the sphere between −1 and a. This is given by the integration of the latitutde angle θ from −π/2 to its value for z=a. That value is such that sin(Θ)=a/1 or Θ=sin⁻¹(a).

The Area of a Truncated Sphere

The infinitesimal of area is the circumference of the circle at θ times dθ.

The radius r of that circle is just cos(θ) so

dA = 2πcos(θ)dθ
and thus
A(Θ) = ∫_−π/2^Θ2πcos(θ)dθ
A(Θ) = 2π∫_−π/2^Θcos(θ)dθ = 2π[sin(θ)]_−π/2^Θ
A(Θ) = 2π[sin(Θ)−(−1)] = 2π[1+sin(Θ)]

Since sin( Θ)=a, the above formula reduces to

A(a) = 2π(1+a)

Hence

∫_SK(p)dA = 2π(1+a)

The integral of the geodesic curvature around the bounding circle was found to be

∫_C k_gds = −2πa

Therefore the contribution of integral of geodesic curvature is positive when a is negative and the truncated sphere is less than half of a sphere. It is negative when a is positive and the truncated sphere is more half of a sphere. It is zero when a is equal to zero which is the case of a hemisphere.

Thus the terms of the Gauss-Bonnet Theorem are

∫_MK(p)dA + ∫_∂M k_gds = 2π(1+a)−2πa
= 2π

A truncated sphere is topologically equivalent to a disk and its Euler-Poincaré characteristic is 1.

Therefore the integral of the Gaussian curvature over the manifold plus the integral of the geodesic curvature over the boundary of the manifold does reduce to 2π times the Euler-Poincaré characteristic of the manifold.

For more on aspects of the Gauss-Bonnet Theorem.