My question involves the symmetries of a (pseudo)-Riemannian metric preserving the Levi-Civita connection (LCC), its unique torsion-free metric connection. For a basic example, one notes that the standard flat connection on $\mathbb R^d$ is the LCC of $\sum_i a_i dx_i\otimes dx_i$ for any non-zero constants $a_i$. This example suggests that sufficiently rigid transformations of a metric may fix the LCC.

**Question.** Let $M$ be a (smooth) manifold. Let $\mathcal G$ be a maximal group of vector bundle automorphisms of $\DeclareMathOperator{Sym}{Sym}\Sym^2(T^*M)$ with the property that the action of $\mathcal G$ on the non-degenerate sections leaves the LCC invariant. What are the geometries of the $\mathcal G$-orbits? Is $\mathcal G$ finite dimensional?