Transverse submanifolds

Let $S$ and $T$ be embedded submanifolds of a smooth manifold $M$ . They are transverse at $p\in S\cap T$ if

T_pS+T_pT=T_pM.

They are transverse, written $S\pitchfork T$ , if this condition holds at every point of $S\cap T$ . Equivalently, the linear map $T_pS\oplus T_pT\to T_pM$ , $(v,w)\mapsto v-w$ , is surjective for each intersection point. Disjoint submanifolds are transverse by convention because there are no intersection points at which the condition can fail.

Transverse intersection theorem

If $S\pitchfork T$ , then $S\cap T$ is an embedded submanifold of $M$ with

T_p(S\cap T)=T_pS\cap T_pT

and codimension $\operatorname{codim}(S\cap T)=\operatorname{codim}S+\operatorname{codim}T$ . Equivalently, its dimension is $\dim S+\dim T-\dim M$ .

Examples and non-examples

The coordinate axes in $\mathbb R^2$ meet transversely at the origin. The $x$ -axis and the parabola $y=x^2$ do not: at their intersection both tangent spaces are the $x$ -axis, so their sum does not fill $\mathbb R^2$ .

Conventions and scope

Transversality is a condition only along the actual intersection. It is stronger than the set-theoretic statement that the intersection has the expected dimension, and it is distinct from orthogonality: no metric is required.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Springer DOI record. Relevant: Chapter 6, transversality and intersections.
V. Guillemin and A. Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. AMS DOI record. Relevant: Chapter 2, transversality and intersection theory.