Clean intersection

Submanifolds $A,B\subseteq M$ intersect cleanly when $A\cap B$ is an embedded submanifold andat every $x\in A\cap B$. A transverse intersection is the special case in which $T_{x}A+T_{x}B=T_{x}M$.

Submanifolds $A,B\subseteq M$ intersect transversely when $T_{x}A+T_{x}B=T_{x}M$ at every intersection point. The transverse intersection theorem then determines the dimension and smoothness of $A\cap B$.