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Three Dimensional Affine Hyperspheres Generated by Two Dimensional Partial Differential Equations
Author(s) -
Vrancken Luc
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200204)237:1<129::aid-mana129>3.0.co;2-o
Subject(s) - mathematics , hypersphere , affine transformation , geodesic , affine connection , connection (principal bundle) , mathematical analysis , affine geometry , affine combination , affine plane (incidence geometry) , affine hull , metric (unit) , partial differential equation , affine coordinate system , pure mathematics , affine space , geometry , operations management , plane curve , economics
Abstract It is well‐known that locally strongly convex affine hyperspheres can be determinedas solutions of differential equations of Monge‐Ampère type. In this paper we study in particular the 3‐dimensional case and we assume that the hypersphere admits a Killing vector field (with respect to the affine metric) whose integral curves are geodesics with respect to both the induced affine connection and the Levi‐Civita connection of the affine metric. We show that besides the already known examples, such hyperspheres can be constructed starting from the 2‐dimensional Poisson equation, the 2‐dimensional sine‐Gordon equation or the 2‐dimensional cosh‐Gordon equation.