Projective vs metric structures

Abstract

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this connection and the metric. The article also provides a basic information on invariants of projective structures, including the treatment via Cartan's normal projective connection. In particular we show that there is a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[nabla]).

References in corpus (4)

• The principle of equivalence and projective structure in space-times
• Metrisability of three-dimensional projective structures
• Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
• Is dark energy meaningless?

Cited by in corpus (12)

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• Einstein metrics in projective geometry
• Projective Compactifications and Einstein metrics
• A criterion for compatibility of conformal and projective structures
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• General structure of Thomas$-$Whitehead gravity
• A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors
• A Projective-to-Conformal Fefferman-Type Construction
• Metrisability of projective surfaces and pseudo-holomorphic curves
• A note on the coincidence of the projective and conformal Weyl tensors
• Weyl Covariant Quadratic Curvature Gravity in 3-Dimensional Riemann-Cartan-Weyl Space-Times