# Metrics with prescribed Levi-Civita connection

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?

• By "leaves the LCC" invariant, do you mean that it leaves $\nabla X$ invariant as an element of $\text{Hom}(TM,TM)$ for each $X\in\Gamma(TM)$? Apr 5 '19 at 11:28
• Yes, meaning that for any fixed vector fields $X,Y$, for any non-degenerate metric $m$, and any element of the automorphism group $h\in \mathcal G$ described above, the torsion-free metric connections $\nabla_m$ for $m$ and $\nabla_{h\cdot m}$ for $h\cdot m$ satisfy $(\nabla_mX)(Y)=(\nabla_{h\cdot m}X)(Y)$. Apr 6 '19 at 20:13