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— We discuss some geometric aspects of PSL(2,C), SL(2,C), and the space G of the geodesics of H3 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.

séminaire de théorie spectrale et géométrie

On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">PSL</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℂ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and on the space of geodesics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℍ</mml:mi> <mml:mn>3</mml:mn> </mml:msup></mml:math> as holomorphic Riemannian manifolds

On PSL(2,ℂ) and on the space of geodesics of ℍ 3 as holomorphic Riemannian manifolds