an:04030543
Zbl 0633.58014
Mercuri, Francesco; Palmieri, Giuliana
Morse theory with low differentiability
EN
Boll. Unione Mat. Ital., VII. Ser., B 1, 621-631 (1987).
00153001
1987
j
58E10 53C22 53C60
closed geodesics; Morse lemma; Finsler manifold
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 \textit{H. H. Matthias} [Bonn. Math. Schr. 126 (1980; Zbl 0481.53042)], by a different technique (finite- dimensional approximation of the space \(\Lambda\) \({\mathcal M})\).
M.Degiovanni
Zbl 0481.53042