# Maximally symmetric affine manifold

As a physicist who knows (something) about General Relativity, I'm accustomed to the term "maximally symmetric space" being an $$n$$-dimensional manifold with $$\frac{n(n+1)}{2}$$ Killing vectors. A demonstration of that can be found in Weinberg's book "Gravitation and Cosmology", but it is always assumed that the manifold is pseudo-Riemannian, and the connection is the Levi-Civita connection.

Question(s): Is the above statement true for affine manifolds? Can you recommend me some bibliography on the subject?

P.D.: I'm interested in considering connections with torsion and non-metricity.

• Careful: affine space has a symmetry group of dimension $n(n+1)$, which is the maximal possible dimension of symmetry group of an affine connection. (See Richard Sharpe's book Differential Geometry for proof). – Ben McKay Nov 22 '18 at 10:05
• For surfaces, you can find a comprehensive discussion of the homogeneous affine surfaces (first classified by Opozda) in arxiv.org/abs/1512.05515. They indeed have to arise from constant curvature metrics just exactly when their affine symmetries have dimension 3 or more. – Ben McKay Nov 22 '18 at 10:14