If $(M,g)$ is a riemannian manifold, then its tangent bundle $TM$ has a natural riemannian metric (as described here).
In short, consider two vectors $V_1,V_2$ tangent to $TM$ at the point $(p,v)$. These two vectors are the derivatives of two curves, say $P_i=(p_i,v_i):I\to TM$ with $P_i(0)= (p,v)$ and $P_i'(0)=V_i$. Now, the components $v_i$ are already vectors (more precisely, vector fields along $p_i$) and, unless $M=\mathbb{R}^n$, we can't derivative them a second time and get another vector tangent to $M$. But we can take covariant derivatives and get vectors tangent to $M$. The natural definition follows: $$\tilde{g}_{(p,v)}(V_1,V_2):= g(p_1'(0),p_2'(0))+g\left( \frac{D v_1}{dt}(0), \frac{Dv_2}{dt}(0)\right).$$
This is a very nice definition because it's coordinate free.
My question is: what about the Levi--Civita connection on $TM$?
I'm working in a specific case (unit tangent bundle of hyperbolic plane $\mathbb{H}^2$) in which I did lots of computation: I found an orthonormal frame, Lie brackets and then found the connection in terms of this frame using Koszul formula. However, it's strongly depending on coordinates.
I tried to organize things but couldn't recognize it as some coordinate free definition, for example in terms of the connection of $M$ and without choosing a specific frame...
For the sphere $\mathbb{S}^2\subset \mathbb{R}^3$ or the hyperboloid model for $\mathbb{H}^2$ in the Lorentz space we have nice descriptions for their connections: extend the vector fields, apply the ambient connection and then just take the tangent components.
Since the unit bundle is a sub manifold of $TM$ we can also think like this, but first we need to know the ambient connection.
Edit: since this seems to be a basic topic on riemannian geometry, I would gladly accept references.
