Center-projective connections

Flat Projective Connections

Complete Projective Connections

The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature. Moreover, I explain the relationship between projective and affine connections.

Projective Connections and the Algebra of Densities

Projective connections first appeared in Cartan’s papers in the 1920’s. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Lapl...

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 sec...

Some Properties of Projective Semi-Symmetric Connections

We investigate the properties of projective semisymmetric connections of a Riemannian manifold. Some interesting results with respect to the invariant of this semi-symmetric connection are given. Mathematics Subject Classification: 53C12