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Uzbekistan Tashkent
Section Mathematics
Title On the group of isometries of foliated manifold
Author(-s) Sharipov A.S.a
Affiliations National University of Uzbekistana
Abstract 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.
Keywords Riemannian manifold, foliation, isometric mapping, foliated manifold, the group of isometries, metric function
UDC 514.3
MSC 53C12, 53C22
DOI 10.20537/vm140110
Received 5 February 2014
Language Russian
Citation 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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