The second fundamental form on a submaniford Given $M$ a Riemannian manifold with metric the levi-civita connection $\nabla$, $N$ embedded submanifold of $M$, and vector fields $X$, $Y$ on $N$, define $f(X,Y)$ to be the orthogonal projection of $\nabla_{\bar Y}\bar X$ onto the orthogonal complement of $TN$, where $\bar X$ and $\bar Y$ are extensions of $N$. 
I wonder whether this is well-defined. First, I have no idea whether it is always possible to extend vector fields from an embedded submanifold. Second, I do not know why the definition is independent of the extensions of $Y$.
FOR THE FIRST QUESTION: I know that it is always possible to extend from a properly embedded submanifold. What about arbitrary embedded submanifold then?
FOR THE SECOND QUESTION: I know that $\nabla$ only cares about local data of $Y$, but here extending from $N$ does not guarantee us local data(there isn't any neighborhood of $M$ contained in $N$, if $N$ has dimension strictly smaller). 
 A: I am not quite sure what you mean with the issue of embedded vs. properly embedded. For an embedded submanifold $N\subset M$, you should always have local charts $(U,u)$ such that $u(U\cap N)$ is the intersection of $u(U)$ with a linear subspace of $\mathbb R^n$. Under these assumptions, you can certainly extend a local vector field on $N$ defined on $U\cap N$ to a local vector field on $M$ defined on $U$, which certainly is sufficient for what you need. 
Concerning the question of being well-defined, the situation is very simple in one variable. The fact that $(\bar X,\bar Y)\mapsto \nabla_{\bar Y}\bar X$ is linear over smooth functions in the second variable easily implies that for any point $x$, the values $\nabla_{\bar Y}\bar X(x)$ depends only on $\bar Y(x)=Y(x)$. In the other variable and additional argument is needed. You have assumed from the beginning that $Y$ and $X$ are tangent to $N$. In particular, this implies that for any local extension $\bar Y$ of $Y$, any flow line of $\bar Y$ that starts in a point of $N$ remains in $N$. Thus you can invoke another well known fact about linear connections (that is used for example to define the covariant derivatives along curves): The value $\nabla_{\bar Y}\bar X(x)$ depends only on the values of $\bar X$ along the flow line of $\bar Y$ through $x$. Since for $x\in N$ that flow line remains in $N$, the restriction of $\bar X$ to the flow line is independent of the extension. 
