SCHWEITZER 1994 Problem 10

Let $F^2$ be a closed, oriented 2-dimensional smooth surface, $f : F^2 \to F^2$ is a smooth homeomorphism whose order is an odd prime p (i.e., the p-th iterate $f \circ f \circ \cdots \circ f$ is the identity). Then f has a finite number of fixed points: $P_1 , ..., P_s$ . In the tangent plane at the fixed point $P_i$ , a positively directed (i.e., compatible with the direction of the surface) base can be chosen in which f is differentiated by a rotation with positive angle $2\pi k_i/p$ , where $k_i$ is a natural number, $0 < k_i < p$ . Prove that $$\sum_{i = 1}^s k_i^{p-2}\equiv0\pmod{p}$$