Projective relatedness in 4-dimensional manifolds of any signature

This talk will consider a 4-dimensional manifold M admitting a metric g of arbitrary signature and with Levi-Civita connection D. Then one allows g' to be any other metric (of arbitrary signature) on M with Levi-Civita connection D' and assumes D and D' to be projectively related, that is, D and D' have identical families of (unparametrised) geodesics. The question then is to establish the relationships between D and D', between g and g' and also between the signatures of g and g'. (It is mentioned here that the situation when (M,g) is an Einstein space is known and this fact will be briefly recapped.) The results to be described were achieved in collaboration with David Lonie and Zhixiang Wang in Aberdeen.

The techniques will involve holonomy theory and the finding of the subalgebras o(4), o(1,3) and o(2,2) (representing g) in a convenient form. The case when g is of Lorentz signature is of particular importance in Einstein's general theory of relativity and the situation for vacuum and cosmological metrics will be briefly discussed. If time permits, some other related remarks on the connection between the symmetries of g and g' will be made and also on (the apparently unrelated problem of) a converse to Weyl's conformal theorem.