Global Bifurcation of Fixed Points and the Poincaré Translation Operator on Manifolds

Let $M$ be a differentiable manifold and $f: [0,\infty)\times M \to M$ be a $C^1$ map satisfying the condition $f(0,p) = p$ for all $p\in M$. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field $w\colon M\to TM$, given by $w(p) = {\partialf \over \partial\lambda} (0,p)$, is well defined and nonzero, then we obtain global bifurcation. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.