I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either assumes $M$ is a submanifold of Euclidean space.
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1$\begingroup$ Why is this assumption a problem, given that any manifold can be embedded in Euclidean space? $\endgroup$ – Nate Eldredge May 6 '17 at 0:43

$\begingroup$ Oh I want M to be Riemannian, so I'm not sure if it can be embedded if it is not compact... Regarless, I can't find a legible (due to super old terminiolody) text with this type of result. $\endgroup$ – AIM_BLB May 6 '17 at 1:05

2$\begingroup$ Every smooth manifold, compact or not, can be embedded in $\mathbb{R}^{n}$ for $n$ large enough. $\endgroup$ – Bilateral May 6 '17 at 1:17

$\begingroup$ Driver, Emery, Hsu? $\endgroup$ – Nate Eldredge May 6 '17 at 1:52

1$\begingroup$ Nevertheless, embedding a Riemannian manifold is a violent act, like choosing a basis for a vector space, since there is no canonical choice. It is even worse than choosing a basis, since there is no algorithm, even if the Riemannian metric is explicitly expressed in coordinates. $\endgroup$ – Ben McKay May 6 '17 at 8:36