Submaximal metric projective and metric affine structures

الأصل · EN

We prove that the next possible dimension after the maximal n²+2n for the Lie algebra of local projective symmetries of a metric on a manifold of dimension n>1 is n²-3n+5 if the signature is Riemannian or n=2, n²-3n+6 if the signature is Lorentzian and n>2, and n²-3n+8 elsewise. We also prove that the Lie algebra of local affine symmetries of a metric has the same submaximal dimensions (after the maximal n²+n) unless the signature is Riemannian and n=3,4, in which case the submaximal dimension is n²-3n+6.