Nontransversal

Nontransversal is a term used in differential topology and geometry to describe a failure of transversality in the intersection of submanifolds or in the interaction of a map with a submanifold. Transversality is a condition that ensures intersections are well-behaved and stable under perturbations; nontransversality indicates a degenerate or special case.

Definition in submanifold intersections: Suppose A and B are submanifolds of a smooth manifold M and they meet at a point x in A ∩ B. The intersection is transverse at x if the sum of their tangent spaces equals the tangent space of M at x, i.e., T_xA + T_xB = T_xM. Equivalently, the dimensions satisfy dim(A) + dim(B) − dim(T_xA ∩ T_xB) = dim(M). Nontransversal means this condition fails: the tangent spaces do not span T_xM, so the intersection can be degenerate, with higher multiplicity or larger tangent contact.

Nontransversality in maps: For a smooth map f: N → M and a submanifold S ⊆ M, f is transverse to S at p if df_p(T_pN) + T_{f(p)}S = T_{f(p)}M. If this fails, the map is nontransverse to S at p, indicating a degenerate interaction or a family of nearby points with the same image.

Consequences and handling: Transverse intersections are stable under small perturbations and often yield submanifolds of predictable dimension. Nontransversal intersections are non-generic and can be perturbed to achieve transversality (Thom transversality theorem), though some settings (such as symplectic or contact geometry) require additional structure, leading to concepts like Morse–Bott or excess intersection.

See also: transversality, intersection theory, Morse theory, Thom’s transversality theorem.