Morse theory with low differentiability. (English) Zbl 0633.58014

Mercuri, Francesco; Palmieri, Giuliana

Boll. Unione Mat. Ital., VII. Ser., B 1, 621-631 (1987).

The authors generalize the Morse lemma to functions of class \({\mathcal C}^ 1\), which are required to be twice differentiable only at the critical points. This result is applied to show that every simply connected compact Finsler manifold \({\mathcal M}\) possesses infinitely many (geometrically distinct) nonconstant closed geodesics, provided that all closed geodesics are nondegenerate and that the Betti numbers \(b_ k\) of the space \(\Lambda\) \({\mathcal M}\) of free loops are not bounded as \(k\to \infty\). The same result was proved by H. H. Matthias [Bonn. Math. Schr. 126 (1980; Zbl 0481.53042)], by a different technique (finite- dimensional approximation of the space \(\Lambda\) \({\mathcal M})\).

Reviewer: M.Degiovanni

MSC:

58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22	Geodesics in global differential geometry
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

closed geodesics; Morse lemma; Finsler manifold