Two closed geodesics on compact simply connected bumpy Finsler manifolds.(English)Zbl 1357.53086

Author’s abstract: We prove the existence of at least two distinct closed geodesics on a compact simply connected manifold $$M$$ with a bumpy and irreversible Finsler metric, when $$H^{\ast }(M;\mathbf Q)\cong T_{d,h+1}(x)$$ for some integer $$h\geq 2$$ and even integer $$d\geq 2$$. Consequently, together with earlier results on $$S^{n}$$, it implies the existence of at least two distinct closed geodesics on every compact simply connected manifold $$M$$ with a bumpy irreversible Finsler metric.

MSC:

 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics) 53C22 Geodesics in global differential geometry 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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