Infinitedimensional Riemannian Manifolds with Uniform Structure
DOI:
https://doi.org/10.20535/1810-0546.2016.4.70018Keywords:
Infinitedimensional space, Riemmanian manifold, Differential geometryAbstract
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.
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.
A.B. Ivanov, “Behavior of bounded solutions of quasilinear elliptic equations on Riemannian manifolds”, Rus. J. Math. Phys., vol. 15, no. 1, pp. 144–144, 2008.
X. Mo, “The existence of harmonic maps from Finsler manifolds to Riemannian manifolds”, Sci. China. Ser. A: Mathematics, vol. 48, no. 1, pp. 115–130, 2005.
E. Grong, “Sub-Riemannian geometry on infinite-dimensional manifolds”, J. Geometric Analysis, vol. 25, no. 4, pp. 2474–2515, 2015.
Yu.V. Bogdansky, “Laplacian with respect to a measure on a Hilbert space and of the Dirichlet problem for the Poisson equation”, Ukrayinskyi Matematychnyi Jurnal, vol. 63, no. 9, pp. 1169–1178, 2011 (in Russian).
S. Leng, An introduction to Differential Manifolds Theory.Moscow,USSR: Mir, 1967 (in Russian).
Yu.L. Daletsky and Ya.I. Belopolskaya, Stochastic Equations and Differential Geometry. Kyiv, Ukraine: Vyscha Shkola, 1989 (in Russian).
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).
Downloads
Published
Issue
Section
License
Copyright (c) 2017 NTUU KPI Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
