Consider a $d$-dimensional Riemannian manifold embedded in Euclidean space $\mathcal{M}\subset \mathbb{R}^N$ endowed with a metric $g$. We are given the Riemannian metric tensor $g_{ij}$ for this manifold. Is it then possible to derive the projector $\Pi_x:\mathbb{R}^N\to\mathcal{T}_x\mathcal{M}$ for each point $x\in\mathcal{M}$ which maps any ambient vector to the tangent space of $x$? If yes, would you be able to provide a way to do so?
Below I give an illustration of this (everything known except $\Pi_x$, $\mathcal{T}_x\mathcal{M}$ and naturally $v\in\mathcal{T}_x\mathcal{M}$):
For each $x\in M$, $T_xM$ can be realized as an linear subvariety of $\mathbb{R}^N$ (a $d$-plane which doesn't necessarily contain the origin), so every point $u\in \mathbb{R}^N$ has an orthogonal projection on $T_x M$, which is the point of $T_x M$ closer to $u$.
This point is obtained considering $u-x$ (the vector joining $x$ and $u$, which is a vector based at $x$) and decomposing as a sum of a vector tangent to $M$ and one orthogonal to it. Then you sum the tangent part to $x$ and obtain the orthogonal projection of $u$.