Locally maximally symmetric space ($\dim {\mathfrak{isom}}_{\mathrm{local}}=n(n+1)/2$)

An $n$-dimensional Riemannian manifold is locally maximally symmetric if its local Killing vector fields attain the dimension bound for Killing vector fields $n(n+1)/2$. This is equivalent to constant sectional curvature. Positive, zero and negative sectional curvature give local spherical geometry, Euclidean geometry and hyperbolic geometry, respectively. Global identifications can reduce the number of globally defined isometries without changing this local property. The spatial slices of an FLRW metric have this property.