Infinitedimensional Riemannian Manifolds with Uniform Structure

Олексій Юрійович Потапенко

Abstract


Background. Solving boundary value problems on infinitedimensional Riemmanian manifolds, in particular re­searching Dirichlet problem, seems to demand for metric completeness. It does not appear to be feasible to state metric completeness in the general case, hence stems the issue of giving sufficient conditions of it.

Objective. Giving sufficient conditions of metric completeness ofinfinitedimensional Riemmanian manifolds and essential examples that would satisfy them.

Methods. Basic results of functional analysis and contemporary differential geometry are used.

Results. Sufficient conditions of infinitedimensional Riemmanian manifolds completeness have been formulated and proved. It has been proved that given conditions are satisfied for by level surfaces of finite codimension with certain bounds on first and second derivatives of the respective functions.

Conclusions. The Sufficient conditions of Riemmanian manifolds completeness – structure uniformity – look to be promising, since they are satisfied for at least by one relatively wide class of surfaces in Hilbert’s space. In terms of future researches, it now appears to be reasonable to devise approaches to considering boundary value problems on such infinitedimensional Riemmanian manifolds.


Keywords


Infinitedimensional space; Riemmanian manifold; Differential geometry

References


D. Gromol et al., Riemannian Geometry in General. Moscow,USSR: Mir, 1971 (in Russian).

M.M. Postnikov, Lectures on Geometry. Semester V: Riemannian Geometry.Moscow,Russia: Factorial, 1998 (in Russian).

O. Calin and D.C. Chang, Geometric Mechanics on Riemannian Manifolds: Applications to Partial Differential Equations. Bos­ton–Basel–Berlin, USA–Switzerland–Germany: Springer Science & Business Media, 2006.

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Yu.V. Bogdansky, “Banach manifolds with a bound structure and the Gauss–Ostrogradsky formula”, Ukrayinskyi Matema­tychnyi Jurnal, vol. 64, no. 10, pp. 1299–1313, 2012 (in Russian).


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DOI: https://doi.org/10.20535/1810-0546.2016.4.70018

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