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The Definition of Topological Manifolds
This article introduces the definition of n-locally Euclidean topological spaces and topological manifolds [13].Riccardi Marco - Via del Pero 102, 54038 Montignoso, ItalyGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665-674, 1991.Zbigniew Karno. The lattice of domains of an extremally disconnected space. Formalized Mathematics, 3(2):143-149, 1992.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En/T. Formalized Mathematics, 12(3):301-306, 2004.John M. Lee. Introduction to Topological Manifolds. Springer-Verlag, New York Berlin Heidelberg, 2000.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93-96, 1991.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Pąk. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009, doi: 10.2478/v10037-009-0024-8.Bartłomiej Skorulski. First-countable, sequential, and Frechet spaces. Formalized Mathematics, 7(1):81-86, 1998.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990
Rotation numbers of invariant manifolds around unstable periodic orbits for the diamagnetic Kepler problem
In this paper, a method to construct topological template in terms of
symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm
the topological template, rotation numbers of invariant manifolds around
unstable periodic orbits in a phase space are taken as an object of comparison.
The rotation numbers are determined from the definition and connected with
symbolic sequences encoding the periodic orbits in a reduced Poincar\'e
section. Only symbolic codes with inverse ordering in the forward mapping can
contribute to the rotation of invariant manifolds around the periodic orbits.
By using symbolic ordering, the reduced Poincar\'e section is constricted along
stable manifolds and a topological template, which preserves the ordering of
forward sequences and can be used to extract the rotation numbers, is
established. The rotation numbers computed from the topological template are
the same as those computed from their original definition.Comment: 8 figures, 1 tabl
Orbifolds, geometric structures and foliations. Applications to harmonic maps
In recent years a lot of attention has been paid to topological spaces which
are a bit more general than smooth manifolds - orbifolds. Orbifolds are
intuitively speaking manifolds with some singularities. The formal definition
is also modelled on that of manifolds, an orbifold is a topological space which
locally is homeomorphic to the orbit space of a finite group acting on .
Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston,
who introduced the term "orbifold". Due to their importance in physics, and in
particular in the string theory, orbifolds have been drawing more and more
attention. In this paper we propose to show that the classical theory of
geometrical structures, easily translates itself to the context of orbifolds
and is closely related to the theory of foliated geometrical structures, cf.
\cite{Wo0}. Finally, we propose a foliated approach to the study of harmonic
maps between Riemannian orbifolds based on our previous research into
transversely harmonic maps
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