I was wondering that what should a restriction of a tensor field on a submanifold mean or look like ? To be more precise, suppose I look at $S^{n}\hookrightarrow \mathbb{R}^{n+1}$ and let's denote the normal vector by $N$. Now suppose $M^{n-1}$ is a smooth embedded hypersurface of $S^n$, with normal vector $N'$. If $X$ is a vector field on $\mathbb{R}^{n+1}$, then it can be written as \begin{equation*} X= Y+fN'+hN \end{equation*} where $Y$ is a vector field on $M$, $f:M\rightarrow \mathbb{R}$ is given by $f=\langle X, N'\rangle$ and $h:M\rightarrow \mathbb{R}$ is given by $h=\langle X, N\rangle$.
Now suppose I start with a tensor field on $\mathbb{R}^{n+1}$, say a $(1,1)$ tensor field $P$ (or an endomorphism of $T\mathbb{R}^{n+1}$ ). Can we have an expression for $P$ as above by writing $P$ as some $(1,1)$ tensor field on $M$ plus some other quantities.
Thanks !