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Section
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Mathematics
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Title
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On the group of isometries of foliated manifold
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Author(-s)
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Sharipov A.S.a
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Affiliations
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National University of Uzbekistana
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Abstract
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The question of the group of isometries of a Riemannian manifold is the main problem of the classical Riemannian geometry. Let $G$ denote the group of isometries of a Riemannian manifold $M$ of dimension $n$ with a Riemannian metric $g$. It is known that the group $G$ with the compact-open topology is a Lie group. This paper discusses the question of the existence of isometric maps of the foliated manifold $(M, F)$. We denote the group of all isometries of the foliated Riemannian manifold $(M, F)$ by $G_{F}$. Studying the structure of the group $G_{F}$ of the foliated manifold $(M, F)$ is a new and interesting problem. First, this problem is considered in the paper of A.Y. Narmanov and the author, where it was shown that the group $G_{F}$ with a compact-open topology is a topological group. We consider the question of the structure of the group $G_{F}$, where $M=R^{n}$ and $F$ is foliation generated by the connected components of the level surfaces of the smooth function $f\colon R^{n}\rightarrow R$. It is proved that the group of isometries of foliated Euclidean space is a subgroup of the isometry group of Euclidean space, if the foliation is generated by the level surfaces of a smooth function, which is not a metric.
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Keywords
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Riemannian manifold, foliation, isometric mapping, foliated manifold, the group of isometries, metric function
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UDC
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514.3
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MSC
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53C12, 53C22
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DOI
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10.20537/vm140110
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Received
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5 February 2014
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Language
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Russian
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Citation
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Sharipov A.S. On the group of isometries of foliated manifold, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 118-122.
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References
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- Tamura I. Topology of foliations: an introduction, American Mathematical Society, 1992, 193 p.
- Narmanov A., Sharipov A. On the group of foliation isometries, Methods of Functional Analysis and Topology, 2009, vol. 15, pp. 195-200.
- Narmanov A., Kaypnazarova G. Metric functions on Riemannian manifolds, Uzbek. Math. J., 2010, no. 2, pp. 113-120.
- Tondeur Ph. Foliations on Riemannian manifolds, New York: Springer-Verlag, 1988.
- O’Neil B. The fundamental equations of a submersion, Michigan Mathematical Journal, 1966, vol. 13, pp. 459-469.
- Myers S.B., Steenrod N. The group of isometrics of a Riemannian manifold, Ann. of Math., 1939, vol. 40, pp. 400-416.
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Full text
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