Metric Connections in Projective Differential Geometry
We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikeš, 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.
Key wordsProjective differential geometry metric connection tractor
AMS(M0S) subject classificationsPrimary 53A20 Secondary 58570
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