I have been trying to define the Levi-Civita connection by directly constructing a Riemannian metric on the cotangent bundle (I know once you have Levi-Civita, this is possible, but I want to do it the other way). I have an idea for a metric on the cotangent bundle, but I cannot prove it is positive definite.
Let $(M,g)$ be a Riemannian manifold. Define $H : T^*M\to\mathbb{R}$ the function $$H(v^{\#}) = \frac{1}{2}||v^{\#}||^2.$$
Then the symplectic structure on $T^*M$ gives rise to a vector field $X_H$ whose flows are the geodesic flow. Define $g'$ a $(2,0)$-tensor on $T^*M$ as $$g'(v, w) = g(D\pi(v), D\pi(w)) + g(D\pi(\mathcal{L}_{X_H}(v)), D\pi(\mathcal{L}_{X_H}(w))),$$ where $\pi : T^*M\to M$ is the projection and $\mathcal{L}$ is the Lie derivative.
Is this $g'$ a Riemannian metric?