Abstract
We study the topology and geometry of the space \(\varUpsilon \) of shapes of abstract tetrahedra, i.e. flat metrics on the 2-sphere with four conical singularities, up to equivalence by orientation-preserving similarities. There is no labeling of the conical singularities; two of them with the same cone-angle can be interchanged by a similarity. We are dealing with the geometric structure introduced by Thurston through the area viewed as a quadratic form. \(\varUpsilon \) is shown to be a quotient of a real analytic fibration over the 3-dimensional space of cone-angles, with each fiber being a thrice-punctured 2-sphere of constant curvature. In particular, \(\varUpsilon \) is shown to be a contractible orbifold of dimension five. It is also observed that the metric given by the area form degenerates transversally to the fibers. A description of the real analytic structure of \(\varUpsilon \) is given by hypergeometric functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 7. Cambridge University Press, Cambridge (1999)
Arnold, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60. Springer, Berlin (1989)
Berger, M.: Geometry II. Universitext. Springer, Berlin (1987)
Boadi, R., Parker, J.: Mostow’s lattices and cone metrics on the sphere. Adv. Geom. 15, 27–53 (2015)
Bonahon, F.: Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots. Stud. Math. Libr. 49, 20 AMS-IAS (2009)
Carathéodory, C.: Theory of Functions of a Complex Variable II. Chelsea Publishing Company, Chelsea (1954)
Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York (1974)
Cooper, D., Hodgson, C.D., Kerckhoff, S.P.: Three-dimensional orbifolds and cone-manifolds. MSJ Mem. Math. Soc. Jpn 5, 220 (2000)
Deligne, P., Mostow, G.G.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. Inst. Hautes Études Sci. 63, 5–89 (1986)
Deraux, M., Falbel, E., Paupert, J.: New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math. 194, 155–201 (2005)
Ghazouani, S., Pirio, L.: Moduli spaces of flat tori with prescribed holonomy. Geom. Funct. Anal. 27, 1289–1366 (2017)
Goldman, W.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (1999)
González, A., López-López, J.L.: Shapes of tetrahedra with prescribed cone angles. Confor. Geom. Dyn. 15, 50–63 (2011)
Klingenberg, W.: Riemannian Geometry. De Gruyter Studies in Mathematics. De Gruyter, Berlin (1995)
Kobayashi, S., Nomizu, K.: Fundations of differential geometry II. Wiley Classics Library. Wiley, New York (1996)
Kojima, S.: Complex hyperbolic cone structures on the configuration spaces. Rend. Ist. Mat. Univ. Trieste 32, 149–163 (2001)
Jones, G., Singerman, D.: Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge University press, Cambridge (1987)
Parker, J.R.: Cone metrics on the sphere and Livné’s lattices. Acta Math. 196, 1–64 (2006)
Pasquinelli, I.: Deligne–Mostow lattices with three fold symmetry and cone metrics on the sphere. Confor. Geom. Dyn. 20, 235–281 (2016)
Pommerenke, C.: Boundary Behavior of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, vol. 299. Springer, Berlin (1992)
Radó, T.: Sur la représentation conforme de domaines variables, Acta Sci. Math. Szeged 1 (1922–1923), 180–186
Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere, The Epstein Birthday Schrift. Geom. Topol. Monogr. 1, 511–549 (1998)
Troyanov, T.: Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324, 793–821 (1991)
Wolf, J.A.: Spaces of Constant Curvature. AMS Chelsea Publishing, Providence (2011)
Acknowledgements
This work was born in 2015 as the result of stimulating conversations between the first author and Lilia Alanís, Manuel Sedano and Xavier Gómez-Mont. The first author is very grateful to all of them. The authors are grateful to the anonymous referee for the careful reading of the paper and for comments that have greatly improved our exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mario Bonk.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by funding from CIC, Universidad Michoacana de San Nicolás de Hidalgo.
Rights and permissions
About this article
Cite this article
González, A., López-López, J.L. Similarity Classes of Tetrahedra. Comput. Methods Funct. Theory 22, 243–260 (2022). https://doi.org/10.1007/s40315-021-00371-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40315-021-00371-y