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A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups

近藤 裕司 広島大学

2022.03.23

概要

In differential geometry, it is one of the central and fundamental problems
to determine whether a given differentiable manifold admits some distinguished
geometric structures or not. Such structures can be, for example, Einstein
or Ricci soliton metrics (cf. [4, 26]) for the setting of Riemannian or pseudoRiemannian manifolds, and K¨ahler-Einstein metrics for K¨ahler manifolds. When
one deals with these problems, it would be natural and useful to add some other
properties, such as homogeneity.
We focus on the problem whether a given Lie group admits distinguished
left-invariant metrics or not, both for the Riemannian and pseudo-Riemannian
cases. Left-invariant metrics on Lie groups have supplied many examples of
distinguished metrics, and have been studied actively. For example, we refer to
[1, 5, 15, 17, 20, 21, 22, 27] and references therein. In particular, we mention
that the Alekseevskii’s conjecture has been recently proved in [2], which had
been an open problem on homogeneous Einstein manifolds with negative scalar
curvature. However, even if we consider the Riemannian cases, the present state
is far from the complete.
If one can classify left-invariant metrics on a given Lie group, then it would
be useful to determine the existence and non-existence of distinguished metrics.
*This work was partly supported by Osaka City University Advanced Mathematical Institute
(MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics). This work
was supported by the Research Institute for Mathematical Sciences, an International Joint
Usage/Research Center located in Kyoto University.
2020 Mathematics Subject Classification. Primary 53C30; Secondary 53C50.
Key words. left-invariant metrics on Lie groups, pseudo-Riemannian metrics, Heisenberg
group, parabolic subgroups, pseudo-Riemannian symmetric spaces. ...

参考文献

[1] F. Barnet, On Lie groups that admit left-invariant Lorentz metrics of constant sectional

curvature, Illinois J. Math. 33 (1989), no. 4, 631–642.

[2] C. B¨

ohm and R. A. Lafuente, Non-compact Einstein manifolds with symmetry,

arXiv:2107.04210v1.

[3] M. Boucetta and O. Tibssirte, On Einstein Lorentzian nilpotent Lie groups, J. Pure Appl.

Algebra 224 (2020), no. 12, 106443, 22 pp.

[4] H.-D. Cao, Recent progress on Ricci solitons, Adv. Lect. Math. 11 (2010), 1–38.

[5] D. Conti and F. A. Rossi, Ricci-flat and Einstein pseudoriemannian nilmanifolds, Complex

Manifolds 6 (2019), no. 1, 170–193.

[6] L. A. Cordero and P. E. Parker, Left-invariant Lorentzian metrics on 3-dimensional Lie

groups, Rend. Mat. Serie VII 17 (1997), 129–155.

[7] K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and

applications, Mathematics and its Applications, 364. Kluwer Academic Publishers Group,

Dordrecht, (1996).

[8] M. Guediri, Sur la compl´

etude des pseudo-m´

etriques invariantes a gauche sur les groupes

de Lie nilpotens, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 4, 371–376.

[9] M. Guediri and M. Bin-Asfour, Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups, Arch. Math. (Brno) 50 (2014), no. 3, 171–192.

[10] K. Y. Ha and J. B. Lee, Left invariant metrics and curvatures on simply connected threedimensional Lie groups, Math. Nachr. 282 (2009), no. 6, 868–898.

[11] T. Hashinaga and H. Tamaru, Three-dimensional solvsolitons and the minimality of the

corresponding submanifolds, Internat. J. Math. 28 (2017), 1750048 (31 pages).

[12] A. Kubo, K. Onda, Y. Taketomi and H. Tamaru, On the moduli spaces of left-invariant

pseudo-Riemannian metrics on Lie groups, Hiroshima Math. J. 46 (2016), 357–374.

[13] Y. Kondo and H. Tamaru, A classification of left-invariant Lorentzian metrics on some

nilpotent Lie groups, Tohoku Math. J. to appear, arXiv:2011.09118v1.

[14] H. Kodama, A. Takahara and H. Tamaru, The space of left-invariant metrics on a Lie

group up to isometry and scaling, Manuscripta Math. 135 (2011), 229–243.

[15] J. Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715–733.

[16] J. Lauret, Degenerations of Lie algebras and geometry of Lie groups, Differential Geom.

Appl. 18 (2003), no. 2, 177–194.

[17] J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011), 1–21.

[18] T. Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups,

Hiroshima Math. J. 12 (1982), 307–320.

[19] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic

subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331–357.

[20] J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976),

no. 3, 293–329.

[21] K. Nomizu, Left-invariant Lorentz metrics on Lie groups, Osaka J. Math. 16 (1979),

143–150.

[22] K. Onda, Examples of algebraic Ricci solitons in the pseudo-Riemannian case, Acta Math.

Hungar. 144 (2014), no. 1, 247–265.

A classification of left-invariant pseudo-Riemannian metrics

21

[23] B. O’Neill, Semi-Riemannian geometry with applications to relativity, Pure and Applied

Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New

York, (1983).

[24] S. Rahmani, M´

etriques de Lorentz sur les groupes de Lie unimodulaires, de dimension

trois, J. Geom. Phys. 9 (1992), no. 3, 295–302.

[25] N. Rahmani and S. Rahmani, Lorentzian geometry of the Heisenberg group, Geom. Dedicata 118 (2006), 133–140.

[26] P. Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series,

vol. 325, Cambridge University Press, Cambridge, 2006.

[27] C. Will, The space of solvsolitons in low dimensions, Ann. Glob. Anal. Geom. 40 (2011),

no. 3, 291–309.

[28] J. A. Wolf, Finiteness of orbit structure for real flag manifolds, Geometriae Dedicata 3

(1974), 377–384.

Yuji Kondo

Department of Mathematics

Graduate School of Science

Hiroshima University

Higashi-Hiroshima 739-8526 JAPAN

E-mail : yuji-kondo@hiroshima-u.ac.jp

...

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