Question about "Stochastic Analysis on Manifolds"

Question

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ dX_s,$$ where $P\left(x\right):\mathbb{R}^N\to T_xM$ is the orthogonal projection operator. The integral is understood to be a Stratonovich integral. He then expands on this and says that $$dX_t=P_{\alpha}\left(X_t\right)\circ dX^{\alpha}_t,$$ where $\alpha$ represents a summation over $1$ to $N$. My question is what vectors are $P\left(X_s\right)$ and $P_{\alpha}\left(X_t\right)$ actually projecting. If anyone has access to the book and knows please let me know.