If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \mathbb R$ is the Itō stochastic integral, then the Itō isometry is the name given to the equality
$$ \| I_\alpha \| _{L^2 (\mathcal C, w)} ^2 = \int _{\mathcal C} \int _0 ^1 \| \alpha _{c(t)} \| ^2 \ \mathrm d t \ \mathrm d w (c) \ .$$
Does a similar relation exist if $\mathbb R^n$ is replaced by an arbitrary connected Riemannian manifold $M$? (Or, if not arbitrary, then at least subjected to some conditions?)
I am looking for something like
$$ \| I_\alpha \| _{L^2 (\mathcal C, w)} ^2 \le C \int _{\mathcal C} \int _0 ^1 \| \alpha _{c(t)} \| ^2 \ \mathrm d t \ \mathrm d w (c) \ ,$$
with $C>0$ depending on the manifold (notice that the equality has become an inequality).
