Previous |  Up |  Next


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:
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


Files Size Format View
ActaOstrav_23-2015-2_1.pdf 488.8Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo