Projective versus metric structures

We present a number of conditions which are necessary for an nn-dimensional projective structure (M,[∇])(M,[∇]) to include the Levi-Civita connection ∇∇ of some metric on MM. We provide an algorithm, which effectively checks whether a Levi-Civita connection is in the projective class and, which finds this connection and the metric, when it is possible. The article also provides basic information on invariants of projective structures, including the treatment via the Cartan normal projective connection. In particular we show that there are 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,[∇])(M,[∇]).