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 Title: The Morse-Sard-Brown Theorem for Functionals on Bounded Fréchet-Finsler Manifolds (English) Author: Eftekharinasab, Kaveh Language: English Journal: Communications in Mathematics ISSN: 1804-1388 Volume: 23 Issue: 2 Year: 2015 Pages: 101-112 Summary lang: English . Category: math . 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 $\mathbb {R}$. (English) Keyword: Fréchet manifolds Keyword: condition (CV) Keyword: Finsler structures Keyword: Fredholm vector fields MSC: 58B15 MSC: 58B20 MSC: 58K05 idZBL: Zbl 1338.58027 idMR: MR3436678 . Date available: 2016-01-19T13:46:11Z Last updated: 2018-01-10 Stable URL: http://hdl.handle.net/10338.dmlcz/144798 . Reference: [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 Reference: [2] Dodson, C.T.J.: Some recent work in Fréchet geometry.Balkan J. Geometry and Its Applications, 17, 2, 2012, 6-21, Zbl 1286.58004, MR 2911963 Reference: [3] Eftekharinasab, K.: Sard's theorem for mappings between Fréchet manifolds.Ukrainian Math. J., 62, 12, 2011, 1896-1905, MR 2958816, 10.1007/s11253-011-0478-z Reference: [4] Eftekharinasab, K.: Geometry of Bounded Fréchet Manifolds.Rocky Mountain J. Math., to appear, Reference: [5] Eliasson, H.: Geometry of manifolds of maps.J. Differential Geometry, 1, 1967, 169-194, Zbl 0163.43901, MR 0226681, 10.4310/jdg/1214427887 Reference: [6] Glöckner, H.: Implicit functions from topological vector spaces in the presence of metric estimates.preprint, Arxiv:math/6612673, 2006, MR 2269430 Reference: [7] Hamilton, R.S.: The inverse function theorem of Nash and Moser.Bulletin of the AMS, 7, 1982, 65-222, Zbl 0499.58003, MR 0656198, 10.1090/S0273-0979-1982-15004-2 Reference: [8] Müller, O.: A metric approach to Fréchet geometry.Journal of Geometry and Physics, 58, 11, 2008, 1477-1500, Zbl 1155.58002, MR 2463806, 10.1016/j.geomphys.2008.06.004 Reference: [9] Palais, R.S.: Lusternik-Schnirelman theory on Banach manifolds.Topology, 5, 2, 1966, 115-132, Zbl 0143.35203, MR 0259955, 10.1016/0040-9383(66)90013-9 Reference: [10] Palais, R.S.: Critical point theory and the minimax principle.Proc. Symp. Pur. Math., 15, 1970, 185-212, Zbl 0212.28902, MR 0264712, 10.1090/pspum/015/0264712 Reference: [11] Tromba, A.J.: A general approach to Morse theory.J. Differential Geometry, 12, 1, 1977, 47-85, Lehigh University, Zbl 0344.58012, MR 0464304 Reference: [12] Tromba, A.J.: The Morse-Sard-Brown theorem for functionals and the problem of Plateau.Amer. J. Math., 99, 1977, 1251-1256, Zbl 0373.58003, MR 0464285, 10.2307/2374024 Reference: [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, Zbl 0383.58001, MR 0493919, 10.1016/0001-8708(78)90061-0 .

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