Transitive riemannian isometry groups with nilpotent radicals

C. Gordon

Annales de l'institut Fourier (1981)

  • Volume: 31, Issue: 2, page 193-204
  • ISSN: 0373-0956

Abstract

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Given that a connected Lie group G with nilpotent radical acts transitively by isometries on a connected Riemannian manifold M , the structure of the full connected isometry group A of M and the imbedding of G in A are described. In particular, if G equals its derived subgroup and its Levi factors are of noncompact type, then G is normal in A . In the special case of a simply transitive action of G on M , a transitive normal subgroup G ' of A is constructed with dim G ' = dim G and a sufficient condition is given for local isomorphism of G ' and G .

How to cite

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Gordon, C.. "Transitive riemannian isometry groups with nilpotent radicals." Annales de l'institut Fourier 31.2 (1981): 193-204. <http://eudml.org/doc/74496>.

@article{Gordon1981,
abstract = {Given that a connected Lie group $G$ with nilpotent radical acts transitively by isometries on a connected Riemannian manifold $M$, the structure of the full connected isometry group $A$ of $M$ and the imbedding of $G$ in $A$ are described. In particular, if $G$ equals its derived subgroup and its Levi factors are of noncompact type, then $G$ is normal in $A$. In the special case of a simply transitive action of $G$ on $M$, a transitive normal subgroup $G^\{\prime \}$ of $A$ is constructed with $\dim G^\{\prime \} = \dim G$ and a sufficient condition is given for local isomorphism of $G^\{\prime \}$ and $G$.},
author = {Gordon, C.},
journal = {Annales de l'institut Fourier},
keywords = {homogeneous Riemannian manifolds; isometry groups; transitive subgroups; Levi decompositions},
language = {eng},
number = {2},
pages = {193-204},
publisher = {Association des Annales de l'Institut Fourier},
title = {Transitive riemannian isometry groups with nilpotent radicals},
url = {http://eudml.org/doc/74496},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Gordon, C.
TI - Transitive riemannian isometry groups with nilpotent radicals
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 2
SP - 193
EP - 204
AB - Given that a connected Lie group $G$ with nilpotent radical acts transitively by isometries on a connected Riemannian manifold $M$, the structure of the full connected isometry group $A$ of $M$ and the imbedding of $G$ in $A$ are described. In particular, if $G$ equals its derived subgroup and its Levi factors are of noncompact type, then $G$ is normal in $A$. In the special case of a simply transitive action of $G$ on $M$, a transitive normal subgroup $G^{\prime }$ of $A$ is constructed with $\dim G^{\prime } = \dim G$ and a sufficient condition is given for local isomorphism of $G^{\prime }$ and $G$.
LA - eng
KW - homogeneous Riemannian manifolds; isometry groups; transitive subgroups; Levi decompositions
UR - http://eudml.org/doc/74496
ER -

References

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  1. [1] R. AZENCOTT and E. N. WILSON, Homogeneous manifolds with negative curvature, Part I, Trans. Amer. Math. Soc., 215 (1976), 323-362. Zbl0293.53017MR52 #15308
  2. [2] R. AZENCOTT and E. N. WILSON, Homogeneous manifolds with negative curvature, Part II, Mem. Amer. Math. Soc., 8 (1976). Zbl0355.53026MR54 #13951
  3. [3] C. GORDON, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann., 248 (1980), 185-192. Zbl0412.53026MR81e:53030
  4. [4] S. HELGASON, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York, 1978. Zbl0451.53038
  5. [5] N. JACOBSON, Lie algebras, Wiley Interscience, New York, 1962. Zbl0121.27504
  6. [6] A. L. ONIŠČIK, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl., 50 (1966), 5-58. Zbl0207.33604
  7. [7] H. OZEKI, On a transitive transformation group of a compact group manifold, Osaka J. Math., 14 (1977), 519-531. Zbl0382.57020MR57 #1362
  8. [8] E. N. WILSON, Isometry groups on homogeneous nilmanifolds, to appear in Geometriae Dedicata. Zbl0489.53045

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