Some results on curvature and topology of Finsler manifolds
We investigate the curvature and topology of Finsler manifolds, mainly the growth of the fundamental group. By choosing a new counting function for the fundamental group that does not rely on the generators, we are able to discuss the topic in a more general case, namely, we do not demand that the manifold is compact or the fundamental group is finitely generated. Among other things, we prove that the fundamental group of a forward complete and noncompact Finsler $n$-manifold $(M,F)$ with nonnegative Ricci curvature and finite uniformity constant has polynomial growth of order $\le n-1$, and the first Betti number satisfies $b_1(M)\le n-1$. We also obtain some sufficient conditions to ensure that the fundamental group is finite or is trivial. Most of the results are new even for Riemannian manifolds.