Rocky Mountain Journal of Mathematics

Geometry of bounded Fréchet manifolds

Kaveh Eftekharinasab

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In this paper, we develop the geometry of bounded Fr\'{e}chet manifolds. We prove that a bounded Fr\'{e}chet tangent bundle admits a vector bundle structure. But, the second order tangent bundle $T^2M$ of a bounded Fr\'{e}chet manifold $M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As an application, we prove the existence and uniqueness of an integral curve of a vector field on $M$.

Article information

Rocky Mountain J. Math., Volume 46, Number 3 (2016), 895-913.

First available in Project Euclid: 7 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58A05: Differentiable manifolds, foundations 58B25: Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05]
Secondary: 37C10: Vector fields, flows, ordinary differential equations

Bounded Fréchet manifold second order tangent bundle connection vector field


Eftekharinasab, Kaveh. Geometry of bounded Fréchet manifolds. Rocky Mountain J. Math. 46 (2016), no. 3, 895--913. doi:10.1216/RMJ-2016-46-3-895.

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  • M. Aghasi, C.T.J. Dodson, G.N. Galanis and A. Suri, Infinite dimensional second order ordinary differential equations via $T^2M$, Nonlin. Anal. 67 (2007), 2829–2838.
  • M. Aghasi, A.R. Bahari, C.T.J. Dodson, G.N. Galanis and A. Suri, Second order structures for sprays and connections on Fréchet manifolds, http://, 2008.
  • C.T.J. Dodson, Some recent work in Fréchet geometry, Balkan J. Geom. Appl. 17 (2012), 6–21.
  • C.T.J. Dodson and G.N. Galanis, Second order tangent bundles of infinite dimensional manifolds, J. Geom. Phys. 52 (2004), 127–136.
  • P. Dombrowski, On the geometry of the tangent bundle, J. reine angew. Math. 210 (1962), 73–88.
  • K. Eftekharinasab, Sard's theorem for mappings between Fréchet manifolds, Ukrainian Math. J. 62 (2010), 1634–1641.
  • P. Flaschel and W. Klingenberg, Riemannsche Hilbert-mannigfaltigkeiten. Periodische Geodätische, Lect. Notes Math. 282, Springer Verlag, Berlin, 1972.
  • R.J. Fisher and H.T. Laquer, Second order tangent vectors in Riemannian geometry, J. Korean Math. Soc. 36 (1999), 959–1008.
  • G.N. Galanis, Differential and geometric structure for the tangent bundle of a projective limit manifold, Rend. Sem. Mat. Padova 112 (2004), 103–115.
  • H. Glöckner, Implicit functions from topological vector spaces in the presence of metric estimates., 2006.
  • R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65–222.
  • S. Kobayashi, Theory of connection, Ann. Mat. Pura Appl. 43 (1957), 119–194.
  • K. Kriegl and P.W. Michor, The convenient setting of global analysis, Math. Surv. Mono. 53, American Mathematical Society 1997.
  • S. Lang, Differential and Riemannian manifolds, Grad. Texts Math. 160, Springer-Verlag, New York, 1995.
  • P. Libermann and C.M. Marle, Symplectic geometry and analytical mechanics, D. Reidel Publishing Company, Dordrecht, 1987.
  • P.W. Michor, Manifolds of differentiable mappings, Shiva Math. 3, Shiva Publishing, Orpington, UK, 1980.
  • O. Müller, A metric approach to Fréchet geometry, J. Geom. Phys. 58 (2008), 1477–1500.
  • E. Vassiliou, Transformations of linear connections, Per. Math. Hungar. 13 (1982), 289–308.
  • J. Vilms, Connections on tangent bundles, J. Diff. Geom. 1 (1967), 235–243.
  • K. Yano and S. Ishihara, Differential geometry of tangent bundles of order two, Kodai Math. Sem. 20 (1968), 318–354.