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.
Highlights
Solving boundary value problems on infinitedimensional Riemmanian manifolds, in particular researching Dirichlet problem, seems to demand for metric completeness
It has been proved that given conditions are satisfied for
возникает вопрос приведения ее достаточных условий
Summary
Solving boundary value problems on infinitedimensional Riemmanian manifolds, in particular researching Dirichlet problem, seems to demand for metric completeness. Giving sufficient conditions of metric completeness of infinitedimensional Riemmanian manifolds and essential examples that would satisfy them. Sufficient conditions of infinitedimensional Riemmanian manifolds completeness have been formulated and proved. 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 appears to be reasonable to devise approaches to considering boundary value problems on such infinitedimensional Riemmanian manifolds. Скінченновимірним рімановим многовидам та рімановій геометрії в цілому присвячені, наприклад, праці [1, 2]. Присвячених скінченновимірним рімановим многовидам, можна віднести праці [3,4,5]. Що наведене в [6] узагальнення скінченновимірних ріманових многовидів на нескінченновимірний випадок відрізняється від підходу нашої роботи: в [6] пропонується запровадження ріманового тензора лише на деякому підрозшаруванні дотичного розшарування
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More From: Research Bulletin of the National Technical University of Ukraine "Kyiv Polytechnic Institute"
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