Splitting lemmas for the Finsler energy functional on the space ofH1-curves

We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of H 1 -curves around a critical point or a critical R 1 orbit of a Finsler isometry-invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll–Meyer's splitting lemmas on Riemannian manifolds [Gromoll and Meyer, ‘On differentiable functions with isolated critical points’, Topology 8 (1969) 361–369; Gromoll and Meyer ‘Periodic geodesics on compact Riemannian manifolds’, J. Differential Geom. 3 (1969) 493–510]. As an application, we extend to Finsler manifolds a result by Grove and Tanaka [‘On the number of invariant closed geodesics’, Acta Math. 140 (1978) 33–48; Tanaka, ‘On the existence of infinitely many isometry-invariant geodesics’, J. Differential Geom. 17 (1982) 171–184] about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.