Geodesic
On the group of isometries of foliated manifold
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2014), pp. 118-122
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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\to 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, isometric mapping, foliated manifold, the group of isometries, metric function.
Mots-clés : foliation 
@article{VUU_2014_1_a9,
     author = {A. S. Sharipov},
     title = {On the group of isometries of foliated manifold},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {118--122},
     year = {2014},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2014_1_a9/}
}
TY  - JOUR
AU  - A. S. Sharipov
TI  - On the group of isometries of foliated manifold
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2014
SP  - 118
EP  - 122
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VUU_2014_1_a9/
LA  - ru
ID  - VUU_2014_1_a9
ER  - 
%0 Journal Article
%A A. S. Sharipov
%T On the group of isometries of foliated manifold
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2014
%P 118-122
%N 1
%U http://geodesic.mathdoc.fr/item/VUU_2014_1_a9/
%G ru
%F VUU_2014_1_a9
A. S. Sharipov. On the group of isometries of foliated manifold. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2014), pp. 118-122. http://geodesic.mathdoc.fr/item/VUU_2014_1_a9/

[1] Tamura I., Topology of foliations: an introduction, American Mathematical Society, 1992, 193 pp. | MR | MR | Zbl | Zbl

[2] Narmanov A., Sharipov A., “On the group of foliation isometries”, Methods of Functional Analysis and Topology, 15 (2009), 195–200 | MR | Zbl

[3] Narmanov A., Kaypnazarova G., “Metric functions on Riemannian manifolds”, Uzbek. Math. J., 2010, no. 2, 113–120 | MR

[4] Tondeur Ph., Foliations on Riemannian manifolds, Springer-Verlag, New York, 1988 | MR | Zbl

[5] O'Neil B., “The fundamental equations of a submersion”, Michigan Mathematical Journal, 13 (1966), 459–469 | DOI | MR | Zbl

[6] Myers S. B., Steenrod N., “The group of isometrics of a Riemannian manifold”, Ann. of Math., 40 (1939), 400–416 | DOI | MR | Zbl