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Submaximal metric projective and metric affine structures

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