In the Wikipedia article Riemannian connection on a surface, I am struggling to interpret the following:
"Since $\mathcal X(M)$ is a submodule of $C^\infty(M,\mathbf E^3)=C^\infty(M)\otimes \mathbf E^3$, the operator $X\otimes I$ is defined on $\mathcal X(M)$, taking values in $C^\infty(M,\mathbf E^3)$.
Let $P$ be the smooth map from $M$ into $M_3(\mathbf R)$ such that $P(p)$ is the orthogonal projection of $\mathbf E^3$ onto the tangent space at $p$. Thus for the unit normal vector $\mathbf{n}_p$ at $p$, uniquely defined up to a sign, and $\mathbf v$ in $\mathbf E^3$, the projection is given by $P(p)(\mathbf v) = \mathbf v - (\mathbf{v} \cdot \mathbf{n}_p)\mathbf{n}_p$
Pointwise multiplication by $P$ gives a $C^\infty(M)$-module map of $C^\infty(M, \mathbf E^3)$ onto $\mathcal{X}(M)$ . The assignment $$ \nabla_X Y = P((X\otimes I)Y). $$ defines an operator $\nabla_X$ on $\mathcal{X}(M)$ called the covariant derivative, satisfying the following properties ..."
In particular, what is this the operator $X \otimes I$? Also, how can I relate this notion of covariant derivative to the one given in Covariant derivative or in do Carmo's "Differential Geometry of Curves and Surfaces"?