Clean intersection - Imported Knowlpedia Content

Definition

Let $S$ and $T$ be embedded submanifolds of a smooth manifold $M$ . They intersect cleanly if $C=S\cap T$ is an embedded submanifold of $M$ and, at every $p\in C$ ,

T_pC=T_pS\cap T_pT.

The first condition controls the set-theoretic intersection, while the second requires its smooth structure to have exactly the tangent directions common to $S$ and $T$ . The definition permits $C$ to be empty and does not require the tangent spaces of $S$ and $T$ to span $T_pM$ .

Relation to transversality

Every pair of transverse submanifolds intersects cleanly by the transverse intersection theorem. Clean intersection is weaker: along a connected component of $C$ , the integer

e=\operatorname{codim}_M S+\operatorname{codim}_M T-\operatorname{codim}_M C

equals the codimension of $T_pS+T_pT$ in $T_pM$ . Thus $e=0$ precisely in the transverse case. Clean intersections and their excess directions are treated in Abraham and Marsden, Chapter 3.

Examples and non-examples

Any embedded submanifold $S\subseteq M$ intersects itself cleanly: the intersection is $S$ , and both sides of the tangent equality are $T_pS$ . Unless $S$ is open in $M$ , this self-intersection is not transverse.

The $x$ -axis and the parabola $y=x^2$ in $\mathbb R^2$ meet only at the origin, but they do not intersect cleanly. Their set-theoretic intersection has zero tangent space, whereas the two curve tangent spaces intersect in the entire $x$ -axis.

Conventions and scope

References

Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea Publishing, 2008. DOI record. Relevant: Chapter 3, clean and transverse intersection conditions.
Victor Guillemin and Shlomo Sternberg, Geometric Asymptotics, American Mathematical Society, 1977. DOI record. Relevant: Chapter I, clean intersections.