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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn068, 44 pages, doi:10.1093/imrn/rnn068 published on July 13, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Higher Schläfli Formulas and Applications. II: Vector-Valued Differential Relations

Jean-Marc Schlenker1 and Rabah Souam2

1 Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse Cedex 9, France
2 Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris Diderot - Paris 7, Géométrie et Dynamique, Site Chevaleret, Case 7012, 75205 - Paris Cedex 13, France

Correspondence: Correspondence to be sent to: schlenker{at}math.ups-tlse.fr

The classical Schläfli formula, and its "higher" analogs given in [23], are relations between the variations of the volumes and "curvatures" of faces of different dimensions of a polyhedra (which can be Euclidean, spherical, or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds.


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