In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of
tangency
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
, and plays a role in
general position. It formalizes the idea of a generic intersection in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.
Definition
Two
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s of a given finite-dimensional
smooth manifold are said to intersect transversally if at every point of
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
, their separate tangent spaces at that point together generate the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
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
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
.
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
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, their intersection is oriented. When the intersection is zero-dimensional, the orientation is simply a plus or minus for each point.
One notation for the transverse intersection of two submanifolds
and
of a given manifold
is
. This notation can be read in two ways: either as “
and
intersect transversally” or as an alternative notation for the set-theoretic intersection
of
and
when that intersection is transverse. In this notation, the definition of transversality reads
:
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 along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. If the maps are
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s, this is equivalent to transversality of submanifolds.
Meaning of transversality for different dimensions
Suppose we have transverse maps
and
where
and
are manifolds with dimensions
and
respectively.
The meaning of transversality differs a lot depending on the relative dimensions of
and
. The relationship between transversality and tangency is clearest when
.
We can consider three separate cases:
#When
, it is impossible for the image of
and
's tangent spaces to span
's tangent space at any point. Thus any intersection between
and
cannot be transverse. However, non-intersecting manifolds vacuously satisfy the condition, so can be said to intersect transversely.
#When
, the image of
and
's tangent spaces must sum directly to
's tangent space at any point of intersection. Their intersection thus consists of isolated signed points, i.e. a zero-dimensional manifold.
#When
this sum needn't be direct. In fact it ''cannot'' be direct if
and
are
immersions at their point of intersection, as happens in the case of embedded submanifolds. If the maps are immersions, the intersection of their images will be a manifold of dimension
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. For example, if manifolds of complementary dimension intersect transversally, the signed sum of the number of their intersection points does not change even if we
isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers ( mass num ...
the manifolds to another transverse intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This descends to a bilinear intersection product on homology classes of any dimension, which is
Poincaré dual to the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
on
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
. Like the cup product, the intersection product is
graded-commutative In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy
:xy = (-1)^ yx,
where , ''x'' , and , ''y'' , d ...
.
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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.
Here is a more specialised example: suppose that
is a
simple Lie group and
is its Lie algebra. By the
Jacobson–Morozov theorem
In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , .
Statement
The statement of Jacobson–Morozov relies on t ...
every nilpotent element
can be included into an
-triple
. The representation theory of
tells us that
. The space
is the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at
to the adjoint orbit
and so the
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
intersects the orbit of
transversally. The space
is known as the "Slodowy slice" after
Peter Slodowy
Peter Slodowy (12 October 1948, in Leverkusen – 19 November 2002, in Bonn) was a German mathematician who worked on singularity theory and algebraic geometry.
He completed his Ph.D. thesis at the University of Regensburg
The University of ...
.
Applications
Optimal control
In fields utilizing the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
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
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.
Smoothness of solution spaces
Using
Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function ...
, whose hypothesis is a special case of the transversality of maps, it can be shown that transverse intersections between submanifolds of a space of complementary dimensions or between submanifolds and maps to a space are themselves smooth submanifolds. For instance, if a smooth
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of an oriented manifold's
tangent bundle—i.e. a
vector field—is viewed as a map from the base to the total space, and intersects the zero-section (viewed either as a map or as a submanifold) transversely, then the zero set of the section—i.e. the singularities of the vector field—forms a smooth 0-dimensional submanifold of the base, i.e. a set of signed points. The signs agree with the indices of the vector field, and thus the sum of the signs—i.e. the fundamental class of the zero set—is equal to the Euler characteristic of the manifold. More generally, for a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
over an oriented smooth closed finite-dimensional manifold, the zero set of a section transverse to the zero section will be a submanifold of the base of codimension equal to the rank of the vector bundle, and its homology class will be
Poincaré dual to the
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle ...
of the bundle.
An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the function, then the zero is simple, i.e. it the graph is transverse to the ''x''-axis at that zero; a zero derivative would mean a horizontal tangent to the curve, which would agree with the tangent space to the ''x''-axis.
For an infinite-dimensional example, the
d-bar operator is a section of a certain
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
bundle over the space of maps from a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
into an
almost-complex manifold. The zero set of this section consists of holomorphic maps. If the d-bar operator can be shown to be transverse to the zero-section, this
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
will be a smooth manifold. These considerations play a fundamental role in the theory of
pseudoholomorphic curves and
Gromov–Witten theory. (Note that for this example, the definition of transversality has to be refined in order to deal with
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
!)
Grammar
"Transversal" is a noun; the adjective is "transverse."
quote from J.H.C. Whitehead, 1959
[ Hirsch (1976), p.66]
See also
*
Transversality theorem
In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. I ...
Notes
References
*
*
*
{{DEFAULTSORT:Transversality (Mathematics)
Differential topology
Calculus of variations
Geometry