Section

Mathematics

Title

On the group of isometries of foliated manifold

Author(s)

Sharipov A.S.^{a}

Affiliations

National University of Uzbekistan^{a}

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 compactopen 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 compactopen 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. 118122.

References

 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. 195200.
 Narmanov A., Kaypnazarova G. Metric functions on Riemannian manifolds, Uzbek. Math. J., 2010, no. 2, pp. 113120.
 Tondeur Ph. Foliations on Riemannian manifolds, New York: SpringerVerlag, 1988.
 O’Neil B. The fundamental equations of a submersion, Michigan Mathematical Journal, 1966, vol. 13, pp. 459469.
 Myers S.B., Steenrod N. The group of isometrics of a Riemannian manifold, Ann. of Math., 1939, vol. 40, pp. 400416.

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