On $\protect \mathrm{PSL}(2,\protect \mathbb{C})$ and on the space of geodesics of $\protect \mathbb{H}^3$ as holomorphic Riemannian manifolds

Christian El Emam

doi:10.5802/tsg.361

On

PSL (2, ℂ)

and on the space of geodesics of

ℍ^{3}

as holomorphic Riemannian manifolds

Christian El Emam ¹

¹ Dipartimento di Matematica Felice Casorati, Universita degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, (Italy)

Séminaire de théorie spectrale et géométrie, Tome 35 (2017-2019), pp. 9-21.

Résumé

We discuss some geometric aspects of $PSL (2, ℂ)$ , $SL (2, ℂ)$ , and the space $𝔾$ of the geodesics of $ℍ^{3}$ equipped with some suitable structures of Riemannian holomorphic manifolds of constant sectional curvature. We also observe that $𝔾$ is a symmetric space for the group $PSL (2, ℂ)$ and use it to deduce some correlations between their holomorphic Riemannian metrics.

Publié le : 2021-04-21

DOI : 10.5802/tsg.361

Affiliations des auteurs :

Christian El Emam ¹

¹ Dipartimento di Matematica Felice Casorati, Universita degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, (Italy)

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     title = {On $\protect \mathrm{PSL}(2,\protect \mathbb{C})$ and on the space of geodesics of $\protect \mathbb{H}^3$ as holomorphic {Riemannian} manifolds},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
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     doi = {10.5802/tsg.361},
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Christian El Emam. On $\protect \mathrm{PSL}(2,\protect \mathbb{C})$ and on the space of geodesics of $\protect \mathbb{H}^3$ as holomorphic Riemannian manifolds. Séminaire de théorie spectrale et géométrie, Tome 35 (2017-2019), pp. 9-21. doi : 10.5802/tsg.361. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.361/

Bibliographie
Cité par

[1] Francesco Bonsante; Christian El Emam On immersions of surfaces into $SL (2, C)$ and geometric consequence (2002) (https://arxiv.org/abs/2002.00810)

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[3] Sorin Dumitrescu; Abdelghani Zeghib Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds, Math. Ann., Volume 345 (2009) no. 1, pp. 53-81 | DOI | MR | Zbl

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[5] Claude R. LeBrun H-space with a cosmological constant, Proc. R. Soc. Lond., Volume 380 (1982), pp. 171-185 | MR | Zbl

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[7] Dong Hoon Lee The structure of complex Lie groups, CRC Research Notes in Mathematics, 429, Chapman & Hall/CRC, 2002 | MR | Zbl

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