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Calculus of generalized hyperbolic tetrahedra

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Abstract

We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.

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References

  1. Alekseevskij, D.V., Vinberg, È.B., Solodovnikov, A.S.: Geometry of Spaces of Constant Curvature. Geometry, II, 1-138, Encyclopaedia Math. Sci., 29, Springer, Berlin (1993)

  2. Chow B., Luo F.: Combinatorial Ricci flows on surfaces. J. Differ. Geom. 63, 97–129 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Cooper D., Rivin I.: Combinatorial scalar curvature and rigidity of ball packings. Math. Res. Lett. 3(1), 51–60 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Fenchel, W.: Elementary geometry in hyperbolic space. In: Bauer, H. de Gruyter Studies in Mathematics 11, Walter de Gruyter & Co., Berlin (1989)

  5. Guo R., Luo F.: Rigidity of polyhedral surfaces, II. Geom. Topol. 13, 1265–1312 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Heard, D.: Computation of hyperbolic structures on 3-dimensional orbifolds. PhD Thesis, Melbourne University, December (2005)

  7. Luo, F.: Rigidity of polyhedral surfaces. J. Differ. Geom. (To appear) arXiv:math.GT/0612714

  8. Luo F.: Volume and angle structures on 3-manifolds. Asian J. Math. 11(4), 555–566 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Luo F.: On Teichmüller space of surfaces with boundary. Duke Math. J. 139(3), 463–482 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Luo F.: 3-dimensional Schläfli formula and its generalization. Commun. Contemp. Math. 10(1), 835–842 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Penner R.C.: The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113(2), 299–339 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Second edition. Graduate Texts in Mathematics, 149. Springer, New York, (2006)

  13. Schlenker J.-M.: A rigidity criterion for non-convex polyhedra. Discret. Comput. Geom. 33(2), 207–221 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Schlenker J.-M.: Small deformations of polygons and polyhedra. Trans. Am. Math. Soc. 359(5), 2155–2189 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Taylor Y.U., Woodward C.T.: 6j symbols for U q (sl 2) and non-Euclidean tetrahedra. Selecta Math. (N.S.) 11(3–4), 539–571 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Thurston, W.: Geometry and Topology of 3-Manifolds. Lecture Notes of Math. Dept. of Princeton University (1978)

  17. Wigner E.P.: Group Theory: And its Application to the Quantum Mechanics of Atomic Spectra Pure and Applied Physics, vol. 5. Academic Press, New York-London (1959)

    Google Scholar 

  18. Yakut A.T., Savas M., Kader S.: On the Schlafli differential formula based on edge lengths of tetrahedron in H 3 and S 3. Geom. Dedicata 138, 99–115 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Ren Guo

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  1. Ren Guo
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Correspondence to Ren Guo.

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Guo, R. Calculus of generalized hyperbolic tetrahedra. Geom Dedicata 153, 139–149 (2011). https://doi.org/10.1007/s10711-010-9561-0

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  • DOI: https://doi.org/10.1007/s10711-010-9561-0

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