Transversally intersecting submanifolds

Definition

Suppose $M$ is a differential manifold and $N,P$ are submanifolds. Then, we say that $N$ and $P$ intersect transversally if we have the following:

• The inclusion map from $P$ to $M$ is a tranversal map to $N$
• The inclusion map from $N$ to $M$ is a transversal map to $P$
• At any point $x$ of $P\cap N$, we have:

$T_{x}M=T_{x}N+T_{x}P$