Proof of the Hopf Umlaufsatz by deformation

The Hopf Umlaufsatz states that the angle of the tangent along a simple smooth closed curve r(t) turns by +360 or -360 degrees. Hopf proved this fact by defining for two parameters s,t the function f(s,t) which gives the angle of the line through r(t) and r(s). This continuously extends to f(t,t), the angle of the tangent at r(t). Hopfs beautiful proof is a prototype for many "index" arguments in differential geometry. [Added Nov 2010: The paper of Hopf]

If one passes from (0,0) to (1,1) along the diagonal, the angle deforms by n times 360 degrees, where n is an integer.
After a deformation of the path, the total change of angle still is n times 360 degrees. A continuous deformation preserves the integer n.
The path is deformed to the boundary curve r(t) = (2t,0) from t=0 to t=1/2 and r(t) = (1,2t-1) from t=1/2 to t=1. Along this path, the total deformation angle is +360 or -360 degrees. It follows that n=1 or n=-1 also along the diagonal.

©, June, 2006