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

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, Volume 35 (2017-2019), pp. 9-21.

Abstract

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.

Published online: 2021-04-21

DOI: 10.5802/tsg.361

Author's affiliations:

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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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, Volume 35 (2017-2019), pp. 9-21. doi : 10.5802/tsg.361. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.361/

References
Cited by

[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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