The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of the boundary and the jump angles at the corners.
More specifically, if 
is any two-dimensional Riemannian
manifold (like a surface in three-space) and if 
is an embedded triangle, then the Gauss-Bonnet formula states
that the integral over the whole triangle of the Gaussian
curvature with respect to area is given by 
minus the sum of the jump angles
minus the integral of the geodesic curvature
over the whole of the boundary of the triangle (with respect to arc
length),
 |
(1)
|
where 
is the Gaussian curvature, 
is the area measure, the 
s are the jump angles of
,
and 
is the geodesic curvature of 
, with 
the arc length measure.
The next most common formulation of the Gauss-Bonnet formula is that for any compact, boundaryless two-dimensional Riemannian manifold,
the integral of the Gaussian curvature over
the entire manifold with respect to area
is 
times the Euler characteristic of the manifold,
 |
(2)
|
This is somewhat surprising because the total Gaussian curvature is differential-geometric in character, but the Euler characteristic is topological in character and does not depend on differential geometry at all. So if you distort the surface and change the curvature at any location, regardless of how you do it, the same total curvature is maintained.
Another way of looking at the Gauss-Bonnet theorem for surfaces in three-space is that the Gauss map of the surface has Brouwer degree given by half the Euler characteristic of the surface
 |
(3)
|
which works only for orientable surfaces where 
is compact. This makes the Gauss-Bonnet theorem
a simple consequence of the Poincaré-Hopf
index theorem, which is a nice way of looking at things if you're a topologist,
but not so nice for a differential geometer. This proof can be found in Guillemin
and Pollack (1974). Millman and Parker (1977) give a standard differential-geometric
proof of the Gauss-Bonnet theorem, and Singer and Thorpe (1996) give a Gauss's
theorema egregium-inspired proof which is entirely intrinsic, without any reference
to the ambient Euclidean space.
A general Gauss-Bonnet formula that takes into account both formulas can also be given. For any compact two-dimensional Riemannian
manifold with corners, the integral of the Gaussian
curvature over the 2-manifold with respect to area is 
times the Euler characteristic
of the manifold minus the sum of the jump
angles and the total geodesic curvature
of the boundary.
