MathDB

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}

10

Problems(1)