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Bi-conformal transformations and the invariant characterisation of conformally separable pseudo-Riemannian manifolds

In this talk I introduce a group of generalised symmetries of a pseudo-Riemannian manifold called "the group of bi-conformal transformations". These transformations act by introducing independent conformal factors on a pair of mutually orthogonal projectors projecting onto non-degenerate subspaces. One can consider continuous groups of bi-conformal transformations and find conditions under which the generating vector fields form a finite dimensional Lie algebra. When this is the case one can compute the maximal dimension of the Lie algebra and the pseudo-Riemannian manifolds which are the maximal spaces. These turn out to be a special case of conformally separable pseudo-Riemannian manifolds called "bi-conformally flat". We are able to give a nice geometric characterisation of bi-conformally flat pseudo-Riemannian manifolds in terms of the vanishing of certain tensors which play a similar role to the Weyl or Cotton tensors for conformally flat pseudo-Riemannian manifolds. To define these tensors we need to introduce a new symmetric connection, the "bi-conformal connection", which has an independent geometric interest. Indeed, with the aid of the bi-conformal connection, we are able to give invariant characterisations of additional types of conformally separable pseudo-Riemannian manifolds in any dimension.
 
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