BLACK-HOLE ENTROPY AND THE DIMENSIONAL CONTINUATION OF THE GAUSS-BONNET THEOREM

The Euclidean black hole has topology R(2) x S-d-2. It is shown that, the Einstein's theory, the deficit angle of a cusp at any point in R(2) and the area of the S-d-2 are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the S-d-2 there. These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension of the most general action yielding second order field equations for the metric in any spacetime dimension is given.