Transversality

Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.

Definition

Two submanifolds of a given finite dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point. Manifolds that do not intersect are vacuously transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the ambient space), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a submanifold whose codimension is equal to the sums of the codimensions of the two manifolds. In the absence of the transversality condition the intersection may fail to be a submanifold, having some sort of singular point.

In particular, this means that transverse submanifolds of complementary dimension intersect in isolated points (i.e., a 0-manifold). If both submanifolds and the ambient manifold are oriented, their intersection is oriented. When the intersection is zero-dimensional, the orientation is simply a plus or minus for each point.

Transversality of maps

The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces at points of intersection of the images generate the entire tangent space of the ambient manifold. If the maps are embeddings, this is equivalent to transversality of submanifolds.

Meaning of transversality for different dimensions

Suppose we have transversal maps $f\_1:\; L\_1\; o\; M$ and $f\_2:\; L\_2\; o\; M$ where $L\_1,\; L\_2\; ext\{\; and\; \}\; M$ are manifolds with dimensions $l\_1,\; l\_2\; ext\{\; and\; \}\; m$ respectively.

The meaning of transversality differs a lot depending on the relative dimensions of $M,\; L\_1$ and $L\_2$. In particular the interpretation of transverse as an opposite of tangential only really makes sense when $l\_1\; +\; l\_2\; =\; m$.

We can consider three separate cases:
#When , it is impossible for the image of $L\_1$ and $L\_2$'s tangent spaces to span $M$'s tangent space at any point. Thus $f\_1$ and $f\_2$ cannot intersect.
#When $l\_1\; +\; l\_2\; =\; m$, the image of $L\_1$ and $L\_2$'s tangent spaces must sum directly to $M$'s tangent space at any point of intersection.
#When $l\_2\; +\; l\_2\; >\; m$ this sum needn't be direct. In fact it "cannot" be direct if $f\_1$ and $f\_2$ are immersions at their point of intersection, as happens in the case of embedded submanifolds.

Intersection product

Given any two smooth submanifolds, it is possible to perturb either of them by an arbitrarily small amount such that the resulting submanifold intersects transversally with the fixed submanifold. Such perturbations do not affect the homology class of the manifolds or of their intersections. Thus, if manifolds of complementary dimension intersect transversally, the signed sum of the number of their intersection points does not change even if we isotope the manifolds to another transverse intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This generalizes to a bilinear intersection product on homology classes of any dimension, which is Poincaré dual to the cup product on cohomology. Like the cup product, the intersection product is graded-commutative.

Examples of transverse intersections

The simplest non-trivial example of transversality is of arcs in a surface. An intersection point between two arcs is transverse if and only if it is not a tangency, i.e., their tangent lines inside the tangent plane to the surface are distinct.

In a three-dimensional space, transverse curves do not intersect. Curves transverse to surfaces intersect in points, and surfaces transverse to each other intersect in curves. Curves that are tangent to a surface at a point (for instance, curves lying on a surface) do not intersect the surface transversally.

Applications

Optimal control

In fields utilizing the calculus of variations or the related Pontryagin maximum principle, the transversality condition is frequently used to control the types of solutions found in optimization problems. For example, it is a necessary condition for solution curves to problems of the form:

:Minimize $int\; F(x,\; y,\; y^prime)\; dx$ where one or both of the endpoints of the curve are not fixed. In many of these problems, the solution satisfies the condition that the solution curve should cross transversally the nullcline or some other curve describing terminal conditions.