Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H_u F \mathcal{M}$ of the orthonormal frame bundle $F \mathcal{M}$ at a frame $u:\mathbb{R}^n\to T_p\mathcal{M}$. Furthermore there are theorems that state that for each curve $\gamma:[0,1]\to\mathcal{M}$ and initial frame $u_0$ there is a unique horizontal lift $\tilde{\gamma}:[0,1]\to F \mathcal{M}$.

**Question 1**

What is the expression for the integrand of the energy functional of a curve $\gamma:[0,1]\to\mathcal{M}$, $$ g_{\gamma}(\dot{\gamma},\dot{\gamma}) $$ in terms of its unique horizontal lift $\tilde{\gamma}:[0,1]\to F \mathcal{M}$? Does the answer depend on $u_0$? Since there is also then a uniquely defined curve in $w:[0,1]\to\mathbb{R}^n$ (which I guess is chart independent), what is the value of $g_{\gamma}(\dot{\gamma},\dot{\gamma})$ via $w$? Could it be just simply $$ g_{\gamma}(\dot{\gamma},\dot{\gamma}) = \langle\dot{w},\dot{w}\rangle_{\mathbb{R}^n} $$?

What is an expression for $\gamma$ in terms of $w$? (I know that $ \dot{\gamma} = \tilde{\gamma}\dot{w}$, but how to solve this for $\gamma$ since there is the lift on the RHS?).

**Question 2**

It seems like in construction stochastic processes on $\mathcal{M}$ the orthonormal frame bundle is heavily used because one can solve stochastic equations in $\mathbb{R}^n$, move them to $F \mathcal{M}$ in a straight forward manner, and then project them down to $\mathcal{M}$. This is, if I understand correctly, the essence of the Eells–Elworthy–Malliavin construction. My question is then, why not define Brownian motion in $\mathcal{M}$ via charts, first down at the Euclidean space and then pulling back to the manifold using the chart? I guess there would be a way to glue the curve back together.

Is the frame bundle really just a chart-free way to talk about the manifold's Euclidean structure in a way that is compatible with the metric?