Transversal map to a submanifold

Definition

Suppose $M$ is a differential manifold and $N$ is a submanifold. Then, if $P$ is a differential manifold and $f : P \to M$ is a smooth map, we say that $f$ is transversal to $N$ if for any point $x \in N$ and $y \in f^{- 1} (n)$ , the composite:

$T_{y} P \to^{D f} T_{x} M \to T_{x} M / T_{x} N$

is surjective. When $N$ is a single point, this is equivalent to saying that that point is a regular value of $f$ .

When the map from $P$ to $M$ is also the inclusion of a submanifold, we say that $P$ and $N$ are transversally intersecting submanifolds of $M$ .