Closed geodesics on Finsler manifolds

Mercuri, Francesco

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 60 (1976) no. 2, pp. 111-117

Résumé

Si descrivono alcuni teoremi sull'esistenza di geodetiche chiuse per varietà di Finsler compatte e semplicemente connesse. Le dimostrazioni appariranno in altri lavori.

Bibliographie

[1] S. I. Alber, On periodicity problems in the calculus of variations in the large, «Amer. Math. Soc. Transl.», 14 (2) (1960), 107-172. | MR

[2] R. Abraham and J. Marsden, Foundation of Mechanics, Benjamin, New York and Amsterdam, 1967. | MR

[3] A. Borel, Seminar on transformation groups, «Ann. of Math. Studies», n. 46, Princeton Unyversity press, Princeton, N.J., 1960. | MR

[4] H. Buseman, The geometry of geodesics, Acad. Press, New York, 1955. | MR | Zbl

[5] H. Buseman and W. Mayer, On the foundation of calculus of variations, «Ann. of Math.», 49 (1941), 173-198. | DOI | MR

[6] W. H. Culter, Negligible subsets of infinite dimensional Frechet manifolds, «Proc. Amer. Math. Soc.», 23 (1969), 668-675. | DOI | MR

[7] H. Elliason, On the geometry of manifolds of maps, «J. of Diff. Geometry», 1 (1967), 156-194. | MR

[8] H. Elliason, Morse theory for closed curves, Symposium in inf. dim. topology, Ann. of Math. Studies, n. 69, Princeton University Press, Princeton, N. J., 1972, 63-77. | MR

[9] A. I. Fet, Variational problems on closed manifolds, «Amer. Math. Soc. Transl.», 90 (1953), 147-206. | MR

[10] D. Gromoll and W. Meyer, Periodic geodesics on compact riemannian manifolds, «J. of Diff. Geometry», 3 (1969), 493-510. | MR | Zbl

[11] W. Klingenberg, The theorem of the three closed geodesics, «Bull. Amer. Math. Soc.», 71 (1965), 605-610. | DOI | MR | Zbl

[12] W. Klingenberg, The space of closed curves on the sphere, «Topology», 7 (1968), 395-415. | DOI | MR | Zbl

[13] W. Klingenberg, Closed geodesics, «Ann. of Math.», 89 (1969), 68-91. | DOI | MR | Zbl

[14] W. Klingenberg, Lecture on closed geodesics (to appear). | MR | Zbl

[15] F. Mercuri, Closed geodesics on Finsler manifolds, «Doctoral diss.», University of Chicago, 1975. | MR

[16] J. Milnor, Morse theory, «Ann. Math. Studies», n. 51, Princeton University Press, Princeton, N. J., 1963. | MR

[17] M. Morse, The calculus of variations in the large, «Amer. Math. Soc. Colloq. Publ.», 18, 1934. | MR | Zbl

[18] R. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340. | DOI | MR | Zbl

[19] R. Palais, Liusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. | DOI | MR | Zbl

[20] R. Palais, Foundation of global non-linear analysis, Benjamin, New York, 1968. | MR | Zbl

[21] R. Palais and S. Smale, A generalized Morse theory, «Bull. Amer. Math. Soc.», 70 (1964), 165-172. | DOI | MR | Zbl

[22] H. Rund, The differential geometry of Finsler spaces, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1959. | MR | Zbl

[23] B. Segre, Geometria non euclidea ed ottica geometrica, I, «Rend Acc. Naz. dei Lincei», serie (8) 7 (1949), 16-19. | MR

[24] E. Spanier, Algebraic topology, McGraw Hill, New York, 1966. | MR

[25] S. Stemberg, Lecture on differential geometry, Prentice Hall, Englewood Cliffs, N. J., 1964. | MR

[26] S. Sullivan, Differential forms and the topology of manifolds, Proceedings of the Japan conference on manifolds, 1973.

[27] D. Sullivan and M. Vigue-Porrier, The homology theory of the closed geodesics problem, Preprint I.H.E.S. | MR

[28] F. Warner, The conjugate locus of a riemannian manifold, «Amer. J. of Math.», 87 (1965), 575-604. | DOI | MR | Zbl