Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
Fréchet manifolds; condition (CV); Finsler structures; Fredholm vector fields
Fréchet manifolds; condition (CV); Finsler structures; Fredholm vector fields
Summary:
In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection $\mathcal {K}$ and if $\xi$ is a smooth Lipschitz-Fredholm vector field on $M$ with respect to $\mathcal {K}$ which satisfies condition (WCV), then, for any smooth functional $l$ on $M$ which is associated to $\xi$, the set of the critical values of $l$ is of first category in $\mathbb{R}$. Therefore, the set of the regular values of $l$ is a residual Baire subset of $\mathbb {R}$.
References:
[1] Bejan, C.L.: Finsler structures on Fréchet bundles. Proc. 3-rd Seminar on Finsler spaces, Univ. Braşov 1984, 1984, 49-54, Societatea de ştiinţe Matematice Romania, Bucharest, MR 0823295
[2] Dodson, C.T.J.: Some recent work in Fréchet geometry. Balkan J. Geometry and Its Applications, 17, 2, 2012, 6-21, MR 2911963 | Zbl 1286.58004
[3] Eftekharinasab, K.: Sard's theorem for mappings between Fréchet manifolds. Ukrainian Math. J., 62, 12, 2011, 1896-1905, DOI 10.1007/s11253-011-0478-z | MR 2958816
[4] Eftekharinasab, K.: Geometry of Bounded Fréchet Manifolds. Rocky Mountain J. Math., to appear,
[5] Eliasson, H.: Geometry of manifolds of maps. J. Differential Geometry, 1, 1967, 169-194, DOI 10.4310/jdg/1214427887 | MR 0226681 | Zbl 0163.43901
[6] Glöckner, H.: Implicit functions from topological vector spaces in the presence of metric estimates. preprint, Arxiv:math/6612673, 2006, MR 2269430
[7] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bulletin of the AMS, 7, 1982, 65-222, DOI 10.1090/S0273-0979-1982-15004-2 | MR 0656198 | Zbl 0499.58003
[8] Müller, O.: A metric approach to Fréchet geometry. Journal of Geometry and Physics, 58, 11, 2008, 1477-1500, DOI 10.1016/j.geomphys.2008.06.004 | MR 2463806 | Zbl 1155.58002
[9] Palais, R.S.: Lusternik-Schnirelman theory on Banach manifolds. Topology, 5, 2, 1966, 115-132, DOI 10.1016/0040-9383(66)90013-9 | MR 0259955 | Zbl 0143.35203
[10] Palais, R.S.: Critical point theory and the minimax principle. Proc. Symp. Pur. Math., 15, 1970, 185-212, DOI 10.1090/pspum/015/0264712 | MR 0264712 | Zbl 0212.28902
[11] Tromba, A.J.: A general approach to Morse theory. J. Differential Geometry, 12, 1, 1977, 47-85, Lehigh University, MR 0464304 | Zbl 0344.58012
[12] Tromba, A.J.: The Morse-Sard-Brown theorem for functionals and the problem of Plateau. Amer. J. Math., 99, 1977, 1251-1256, DOI 10.2307/2374024 | MR 0464285 | Zbl 0373.58003
[13] Tromba, A.J.: The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree. Advances in Mathematics, 28, 2, 1978, 148-173, DOI 10.1016/0001-8708(78)90061-0 | MR 0493919 | Zbl 0383.58001
Search
Partner of
