Relation between different connections on the same manifold As I was thinking about this other problem, I stumbled upon another issue (maybe easier to solve).
Suppose we have got a smooth manifold $M$, endowed with two different metrics ($g_1$,$g_2$). Then we have got two riemannian manifolds $M_1 := (M,g_1)$ and $M_2 := (M,g_2)$.

What is the Levi-Civita connection of $M_2$ when viewed in $M_1$? What is its meaning?

Example: $M_1 = (H^2,dx^2+dy^2)$ and $M_2 = (H^2,\frac{dx^2+dy^2}{y^2})$.
What's the meaning of the connection of $M_2$ in the euclidean space?
 A: This is more of a comment than an answer. I'm not entirely sure what kind of answer you're looking for / I don't necessarily have too much expertise in this area.
In general, I'm not sure if there is necessarily any meaningful relationship between the Levi-Civita connections induced by different metrics on the same smooth manifold. However, when the metrics are pointwise conformally related we have a neat, meaningful relationship. 
Let $M_1=(M,g)$ be a smooth Riemannian manifold. Let $\lambda:M\to\mathbb{R}$ be a smooth function, define $\tilde{g}=e^{2\lambda}g$, and consider the pointwise conformally equivalent Riemannian manifold $M_2=(M,\tilde{g})$. Then we have
$$\nabla^{\tilde{g}}_XY=\nabla^g_XY+(X\lambda)Y+(Y\lambda)X-g(X,Y)\mathrm{grad}(\lambda),$$
where $\nabla^g$ and $\nabla^{\tilde{g}}$ are the Levi-Civita connections of $M_1$ and $M_2$, respectively. This is Exercise 8.5 (has lots of hints) in do Carmo's Riemannian Geometry In terms of usefulness, we can use this formula to compute the Levi-Civita connection for a large class of smooth manifolds easily. I'm not sure what to say about it's "meaning", since that seems very problem specific / it depends on what the manifold and metrics represent.
