A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields

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2008-02-01

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Abstract

We give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie. © 2007 Springer-Verlag.

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10.1007/s00208-007-0158-3

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Bryant, RL, G Manno and VS Matveev (2008). A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields. Mathematische Annalen, 340(2). pp. 437–463. 10.1007/s00208-007-0158-3 Retrieved from https://hdl.handle.net/10161/13154.

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Bryant

Robert Bryant

Phillip Griffiths Distinguished Professor of Mathematics

My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.


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