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Special homogeneous surfaces

Part of: Global differential geometry Analytic and descriptive geometry Special varieties Classical differential geometry

Published online by Cambridge University Press:  22 October 2024

DAVID LINDEMANN  and
ANDREW SWANN
DAVID LINDEMANN
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade 118, Bldg 1530, DK-8000 Aarhus C, Denmark. e-mails: david.lindemann@math.au.dk; david.lindemann@uni-hamburg.de
ANDREW SWANN
Affiliation:
Department of Mathematics and DIGIT, Aarhus University, Ny Munkegade 118, Bldg 1530, DK-8000 Aarhus C, Denmark. e-mail: swann@math.au.dk
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Abstract

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

MSC classification

Primary: 53A15: Affine differential geometry
Secondary: 51N35: Questions of classical algebraic geometry 14M17: Homogeneous spaces and generalizations 53C30: Homogeneous manifolds 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 177 , Issue 2 , September 2024 , pp. 333 - 362
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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