Transversality and preimage manifolds

Let f:M→N be smooth and let Z be an embedded submanifold of N. At a point p mapping into Z, transversality means the image of dfₚ plus the tangent space of Z equals the tangent space of N. No condition is required at points whose images miss Z. If Z is one target point, its tangent space is zero and transversality becomes the regular-value condition. If f is an embedding, it becomes the usual spanning test for two submanifolds intersecting in N.

Choose local coordinates making Z the zero set of its independent normal coordinates. Composing f with those normal coordinates yields a map to a Euclidean space of dimension equal to the codimension of Z. The transversality condition makes zero a regular value of that composition. Therefore the preimage is an embedded submanifold with the same codimension as Z. Its tangent space consists of vectors whose image under df lies in the tangent space of Z, rather than necessarily being zero.

Suppose M has boundary, while N and Z have no boundary. To conclude that the preimage is neat, require both f and its restriction to ∂M to be transverse to Z. Neatness means its boundary is exactly the part lying on the ambient boundary, with the appropriate half-space local model. The restriction test uses tangent vectors to ∂M, not every tangent vector to M. This distinction matters even when the full derivative is surjective at all boundary points.

For example, on the upper half-plane the function f(x,y)=y has a surjective full derivative, but its boundary restriction is constant zero. The inverse image of zero is the boundary line itself; calling the entire line the boundary of a one-dimensional preimage would be wrong. In contrast f(x,y)=x remains a submersion when restricted to the boundary line. Its zero preimage is a vertical half-line whose boundary is the origin. More complicated target boundaries can create corners and require additional hypotheses beyond this version.

Transversality is often achieved by perturbation, but the statement must specify the map family and topology. A finite-dimensional smooth parameterized map transverse to Z gives transverse slices for almost every parameter by applying Sard to a suitable projection; source-boundary applications also need the corresponding boundary-family condition. General smooth-map transversality theorems provide density in suitable smooth topologies. Neither assertion means that an arbitrary perturbation picked without checking its family has the desired property.

A tangency shows why the dimension prediction needs the hypothesis. Two plane curves with parallel tangents can meet in a single exceptional point that splits into two intersections or disappears after a small shift. At transverse intersections, the spanning condition persists under sufficiently small local C¹ changes. Global openness can require compactness and a closed target submanifold so that new problematic intersections do not escape control. Use local derivatives to certify the intersection rather than deciding from a picture that merely appears to cross.