Weyl metrisability for projective surfaces

Details Weyl metrisability for projective surfaces Weyl metrisability for projective surfaces

2009-10-14

arxiv.org/abs/0910.2618v2

Mathematics

Differential Geometry

17 pages, no figures. Version 2 contains a proof of local Weyl metrisability for projective surfaces using EDS theory. The sec

Scientific paper

We show that locally every smooth projective surface M admits a compatible Weyl connection. First this is done using exterior differential system and elliptic PDE theory. Second by showing that the Weyl compatibility problem is equivalent to finding a section with holomorphic image of the `twistor' bundle of conformal inner products over M. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the 2-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

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