Submaximal metric projective and metric affine structures

We prove that the next possible dimension after the maximal n2+2nn2+2n for the Lie algebra of local projective symmetries of a metric on a manifold of dimension n>1n>1 is n2−3n+5n2−3n+5 if the signature is Riemannian or n=2n=2, n2−3n+6n2−3n+6 if the signature is Lorentzian and n>2n>2, and n2−3n+8n2−3n+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+nn2+n) unless the signature is Riemannian and n=3,4n=3,4, in which case the submaximal dimension is n2−3n+6n2−3n+6.