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.
CITATION STYLE
Eastwood, M., & Matveev, V. (2008). Metric Connections in Projective Differential Geometry (pp. 339–350). https://doi.org/10.1007/978-0-387-73831-4_16
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