Submaximal metric projective and metric affine structures

TL;DR

This paper determines the submaximal dimensions of Lie algebras of local projective and affine symmetries of metrics on manifolds, revealing how these dimensions vary with signature and dimension.

Contribution

It establishes the precise submaximal dimensions for local projective and affine symmetry Lie algebras, extending known maximal bounds based on signature and dimension.

Findings

Submaximal dimension for projective symmetries: n^2-3n+5 (Riemannian, n>1)

Submaximal dimension for projective symmetries: n^2-3n+6 (Lorentzian, n>2)

Submaximal dimension for affine symmetries: same as projective, except for Riemannian n=3,4 where it is n^2-3n+6

Abstract

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 $n^2-3n+6$.