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On area and volume in spherical and hyperbolic geometry

Abstract : Our aim is to prove some theorems in hyperbolic geometry based on the methods of Euler, Schubert and Steiner in spherical geometry. These theorems, interesting for their own right, exhibit a way to adapt methods of proofs from spherical to hyperbolic geometry. We give the hyperbolic analogues of trigonometric formulae for right triangles by method of variations and an area formula in terms of sides of triangles, both due to Euler in spherical case. We solve Lexell’s problem which is to find the geometric locus of the third vertices of triangles with a given base and area. This is a joint work with Weixu Su. As next, we give a shorter formula than Euler’s area formula. We use hyperbolic analogues of Cagnoli’s identities to prove two classical geometric results in hyperbolic geometry. Further, we give solutions of Schubert’s and Steiner’s problems: the former is to find the extrema of area for triangles with a fixed base and altitude, the latter is to find the extrema of isoperimetric triangles with a given base. The study of Schubert’s problem is a joint work with Vincent Alberge. Finally, following ideas of Norbert A’Campo, we give the sketch of the proof of Schläfli formula using integral geometry. The mentioned theorems can be generalized to the case of dimension 3 partially by means of the techniques used and developed in this thesis. This leads to an open research area.
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Submitted on : Tuesday, September 11, 2018 - 11:10:15 PM
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  • HAL Id : tel-01872314, version 1

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Elena Frenkel. On area and volume in spherical and hyperbolic geometry. Differential Geometry [math.DG]. IRMA (UMR 7501), 2018. English. ⟨tel-01872314v1⟩

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