Isometry groups of Riemannian solvmanifolds.

*(English)*Zbl 0664.53022A Riemannian solvmanifold is a connected Riemannian manifold M whose isometry group I(M) contains a transitive solvable Lie subgroup. The author constructs a single conjugacy class of almost simply transitive solvable subgroups of I(M) called the subgroups in standard position. Let M be a simply connected solvable Lie group R equipped with a left- invariant Riemannian metric. An algorithm for modifying R into a subgroup S of I(M) in standard position is constructed. If R is unimodular, then S is normal in I(M) which permits to describe the group I(M). The results are used for determining when two given Riemannian solvmanifolds are isometric. For a class of solvmanifolds M, all connected transitive subgroups of I(M) are described.

Reviewer: A.L.Onishchik

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |

##### Keywords:

simply transitive group; Riemannian solvmanifold; isometry group; subgroups in standard position; solvable Lie group
PDF
BibTeX
XML
Cite

\textit{C. S. Gordon} and \textit{E. N. Wilson}, Trans. Am. Math. Soc. 307, No. 1, 245--269 (1988; Zbl 0664.53022)

Full Text:
DOI

**OpenURL**

##### References:

[1] | D. V. Alekseevskiĭ, The conjugacy of polar decompositions of Lie groups, Mat. Sb. (N.S.) 84 (126) (1971), 14 – 26 (Russian). |

[2] | D. V. Alekseevskiĭ, Homogeneous Riemannian spaces of negative curvature, Mat. Sb. (N.S.) 96(138) (1975), 93 – 117, 168 (Russian). |

[3] | Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323 – 362. · Zbl 0293.53017 |

[4] | Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc. 8 (1976), no. 178, iii+102. · Zbl 0355.53026 |

[5] | E. D. Deloff, Naturally reductive metrics and metrics with volume preserving geodesic symmetries on \( NC\) algebras, Thesis, Rutgers Univ., New Brunswick, N. J., 1979. |

[6] | Carolyn Gordon, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann. 248 (1980), no. 2, 185 – 192. · Zbl 0412.53026 |

[7] | Carolyn Gordon, Transitive Riemannian isometry groups with nilpotent radicals, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 2, vi, 193 – 204 (English, with French summary). · Zbl 0441.53034 |

[8] | Carolyn S. Gordon and Edward N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241 – 256. · Zbl 0523.58043 |

[9] | Carolyn S. Gordon and Edward N. Wilson, The fine structure of transitive Riemannian isometry groups. I, Trans. Amer. Math. Soc. 289 (1985), no. 1, 367 – 380. · Zbl 0565.53030 |

[10] | Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038 |

[11] | James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004 |

[12] | Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. · Zbl 0121.27504 |

[13] | Takushiro Ochiai and Tsunero Takahashi, The group of isometries of a left invariant Riemannian metric on a Lie group, Math. Ann. 223 (1976), no. 1, 91 – 96. · Zbl 0318.53042 |

[14] | A. L. Oniscik, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl. (2) 50 (1966), 5-58. · Zbl 0207.33604 |

[15] | Hideki Ozeki, On a transitive transformation group of a compact group manifold, Osaka J. Math. 14 (1977), no. 3, 519 – 531. · Zbl 0382.57020 |

[16] | Edward N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), no. 3, 337 – 346. · Zbl 0489.53045 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.