Metric connections in projective differential geometry

Details Metric connections in projective differential geometry Metric connections in projective differential geometry

2008-06-24

arxiv.org/abs/0806.3998v1

Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006), 339--351, IMA Vol. Math. Appl

Mathematics

Differential Geometry

10 pages

Scientific paper

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.

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