Some results on curvature and topology of Finsler manifolds

Bing Ye Wu¹

¹ Department of Mathematics Minjiang University Fuzhou, Fujiang 350108, China

Annales Polonici Mathematici, Tome 107 (2013) no. 3, pp. 309-320.

Résumé

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.

Détail

DOI : 10.4064/ap107-3-6

Keywords: investigate curvature topology finsler manifolds mainly growth fundamental group choosing counting function fundamental group does rely generators able discuss topic general namely demand manifold compact fundamental group finitely generated among other things prove fundamental group forward complete noncompact finsler n manifold nonnegative ricci curvature finite uniformity constant has polynomial growth order n first betti number satisfies n obtain sufficient conditions ensure fundamental group finite trivial results even riemannian manifolds