## Shapes of tetrahedra with prescribed cone angles

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- by Ahtziri González and Jorge L. López-López PDF
- Conform. Geom. Dyn.
**15**(2011), 50-63 Request permission

## Abstract:

Given real numbers $4\pi >\theta _0\geq \theta _1\geq \theta _2\geq \theta _3>0$ so that $\sum _{j=0}^3\theta _j=4\pi$, we provide a detailed description of the space of flat metrics on the 2-sphere with 4 conical points of cone angles $\theta _0,\theta _1,\theta _2,\theta _3$, endowed with a geometric structure arising from the area function.## References

- A. H. Cruz-Cota,
*The moduli space of hex spheres*, http://arxiv.org/abs/1010.5235, 2010. - P. Deligne and G. D. Mostow,
*Monodromy of hypergeometric functions and nonlattice integral monodromy*, Inst. Hautes Études Sci. Publ. Math.**63**(1986), 5–89. MR**849651**, DOI 10.1007/BF02831622 - F. Fillastre,
*From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston*, http://fillastre.u-cergy.fr/articles.html, 2009. - Herman Gluck, Kenneth Krigelman, and David Singer,
*The converse to the Gauss-Bonnet theorem in PL*, J. Differential Geometry**9**(1974), 601–616. MR**390962** - Misha Gromov,
*Metric structures for Riemannian and non-Riemannian spaces*, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR**1699320** - W. Klingenberg,
*Riemannian geometry*, de Gruyter Stud. Math., vol. 1, de Gruyter, 1982. - Sadayoshi Kojima,
*Complex hyperbolic cone structures on the configuration spaces*, Rend. Istit. Mat. Univ. Trieste**32**(2001), no. suppl. 1, 149–163 (2002). Dedicated to the memory of Marco Reni. MR**1893396** - William P. Thurston,
*Three-dimensional geometry and topology. Vol. 1*, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR**1435975**, DOI 10.1515/9781400865321 - William P. Thurston,
*Shapes of polyhedra and triangulations of the sphere*, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549. MR**1668340**, DOI 10.2140/gtm.1998.1.511 - Marc Troyanov,
*On the moduli space of singular Euclidean surfaces*, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, pp. 507–540. MR**2349679**, DOI 10.4171/029-1/13 - William A. Veech,
*Flat surfaces*, Amer. J. Math.**115**(1993), no. 3, 589–689. MR**1221838**, DOI 10.2307/2375075

## Additional Information

**Ahtziri González**- Affiliation: CIMAT, Mineral de Valenciana, C.P. 36240, Guanajuato, Gto., Mexico
- Email: ahtziri@cimat.mx
**Jorge L. López-López**- Affiliation: Facultad de Ciencias Físico-matemáticas, UMSNH, Ciudad Universitaria, C.P. 58040, Morelia, Mich., Mexico
- Email: jllopez@umich.mx
- Received by editor(s): December 7, 2010
- Published electronically: June 7, 2011
- Additional Notes: The study was partially supported by funding from the UMSNH (by means of a project of the CIC) and the SEP (by means of the Red Temática de Colaboración “Álgebra, topología y análisis”).
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**15**(2011), 50-63 - MSC (2010): Primary 51M20; Secondary 58D17, 51M10, 51M25
- DOI: https://doi.org/10.1090/S1088-4173-2011-00225-6
- MathSciNet review: 2833472