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 e ntropy emerges as the Euler class of a small disk centered at the hori zon multiplied by the area of the S-d-2 there. These results are obtai ned through dimensional continuation of the Gauss-Bonnet theorem. The extension of the most general action yielding second order field equat ions for the metric in any spacetime dimension is given.