Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 811-835, 2019


Pseudo-Riemannian weakly symmetric manifolds of low dimension

Bo Zhang, Zhiqi Chen, Shaoqiang Deng

Received November 10, 2017.   Published online March 26, 2019.

Abstract:  We give a classification of pseudo-Riemannian weakly symmetric manifolds in dimensions $2$ and $3$, based on the algebraic approach of such spaces through the notion of a pseudo-Riemannian weakly symmetric Lie algebra. We also study the general symmetry of reductive $3$-dimensional pseudo-Riemannian weakly symmetric spaces and particularly prove that a $3$-dimensional reductive $2$-fold symmetric pseudo-Riemannian manifold must be globally symmetric.
Keywords:  pseudo-Riemannian manifold; pseudo-Riemannian weakly symmetric manifold; pseudo-Riemannian weakly symmetric Lie algebra; Lorentzian weakly symmetric manifold
Classification MSC:  53C30, 22E46
DOI:  10.21136/CMJ.2019.0515-17


References:
[1] J. Berndt, L. Vanhecke: Geometry of weakly symmetric spaces. J. Math. Soc. Japan 48 (1996), 745-760. DOI 10.2969/jmsj/04840745 | MR 1404821 | Zbl 0877.53027
[2] Z. Chen, J. A. Wolf: Pseudo-Riemannian weakly symmetric manifolds. Ann. Global Anal. Geom. 41 (2012), 381-390. DOI 10.1007/s10455-011-9291-z | MR 2886205 | Zbl 1237.53071
[3] V. del Barco, G. P. Ovando: Isometric actions on pseudo-Riemannian nilmanifolds. Ann. Global Anal. Geom. 45 (2014), 95-110. DOI 10.1007/s10455-013-9389-6 | MR 3165476 | Zbl 1295.53081
[4] S. Deng: An algebraic approach to weakly symmetric Finsler spaces. Can. J. Math. 62 (2010), 52-73. DOI 10.4153/CJM-2010-004-x | MR 2597023 | Zbl 1205.53078
[5] S. Deng: On the symmetry of Riemannian manifolds. J. Reine Angew. Math. 680 (2013), 235-256. DOI 10.1515/crelle.2012.040 | MR 3100956 | Zbl 1273.53047
[6] S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces. Pure and Applied Mathematics 80, Academic Press, New York (1978). MR 0514561 | Zbl 0451.53038
[7] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry. Vol. I. Interscience Publishers, John Wiley & Sons, New York (1963). MR 0152974 | Zbl 0119.37502
[8] A. Selberg: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc., New Ser. 20 (1956), 47-87. MR 0088511 | Zbl 0072.08201
[9] H.-C. Wang: Two-point homogeneous spaces. Ann. Math. 55 (1952), 177-191. DOI 10.2307/1969427 | MR 0047345 | Zbl 0048.40503
[10] J. A. Wolf: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs 142, American Mathematical Society, Providence (2007). DOI 10.1090/surv/142 | MR 2328043 | Zbl 1156.22010
[11] O. S. Yakimova: Weakly symmetric Riemannian manifolds with a reductive isometry group. Sb. Math. 195 (2004), 599-614 (In English. Russian original.); translation from Mat. Sb. 195 (2004), 143-160. DOI 10.1070/SM2004v195n04ABEH000817 | MR 2086668 | Zbl 1078.53043
[12] W. Ziller: Weakly symmetric spaces. Topics in Geometry. In Memory of Joseph D'Atri (Gindikin, S. et al., eds.). Progr. Nonlinear Differ. Equ. Appl. 20, Birkh√§user, Boston (1996), 355-368. DOI 10.1007/978-1-4612-2432-7 | MR 1390324 | Zbl 0860.53030

Affiliations:   Bo Zhang, Zhiqi Chen, Shaoqiang Deng (correspodning author), School of Mathematical Sciences and LPMC, Nankai University, No. 94 Weijin Road, Nankai District, Tianjin 300071, P. R. China, e-mail: zhangbo@mail.nankai.edu.cn, chenzhiqi@nankai.edu.cn, dengsq@nankai.edu.cn


 
PDF available at: