Consider a compact Riemannian manifold $M$ with Riemannian structure given by a metric $g$. Given a chart $(U, (x^i))$ and two vector fields $G_{\alpha}, G_{\beta} \in \Gamma(TM)$ it is possible to define the SDE
$dX_t=G_\alpha(X_s) dt + G_\beta(X_s) dW_t$
where $W_t$ is a standard Brownian motion on the tangent space and $X_t \in TM$. All the references that i looked at use the Stratonovich stochastic integral in which the Ito formula doesn't contains the diffusion term. Is it true that for the Ito integral the ito formula is
$f(X_t)=f(X_0)+ \int_0^t \left(G_\alpha+ \frac{1}{2} \nabla_{G_\beta}G_\beta \right) f(X_s) ds + \int_0^t G_\beta dW_t $
If this is correct, given a frame $\left(\frac{\partial}{\partial x^i}\right)$ and in $U$ the formula above became
$df(X_t)= \left(G_{\alpha}^i \frac{\partial}{\partial x^i} + G_{\beta}^iG_{\beta}^i\frac{\partial^2}{\partial x^i \partial x^i}\right)f(X_t)dt + G_{\beta}^i\frac{\partial}{\partial x^i}f(X_t) dW_t $
is there a way to make this formula tensorial while preserving the martigale property?