HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, transversality 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 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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
s of a given finite-dimensional
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
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, their ...
, their separate tangent spaces at that point together generate the
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
of the ambient manifold at that point. Manifolds that do not intersect are
vacuously In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a s ...
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. 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. One notation for the transverse intersection of two submanifolds L_1 and L_2 of a given manifold M is L_ \pitchfork L_. This notation can be read in two ways: either as “L_1 and L_2 intersect transversally” or as an alternative notation for the set-theoretic intersection L_1\cap L_2 of L_1 and L_2 when that intersection is transverse. In this notation, the definition of transversality reads :L_ \pitchfork L_ \iff \forall p \in L_ \cap L_, T_ M = T_ L_ \oplus T_ L_.


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 (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
s, this is equivalent to transversality of submanifolds.


Meaning of transversality for different dimensions

Suppose we have transverse maps f_1: L_1 \to M and f_2: L_2 \to M where L_1, L_2 and M are manifolds with dimensions \ell_1, \ell_2 and m respectively. The meaning of transversality differs a lot depending on the relative dimensions of M, L_1 and L_2. The relationship between transversality and tangency is clearest when \ell_1 + \ell_2 = m . We can consider three separate cases: #When \ell_1 + \ell_2 < m , 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 any intersection between f_1 and f_2 cannot be transverse. However, non-intersecting manifolds vacuously satisfy the condition, so can be said to intersect transversely. #When \ell_1 + \ell_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. Their intersection thus consists of isolated signed points, i.e. a zero-dimensional manifold. #When \ell_1 + \ell_2 > m this sum needn't be direct. In fact it ''cannot'' be direct if f_1 and f_2 are
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux * ''Immersion'' (album), the third album by Australian group Pendulum * ''Immersion'' (film), a 2021 ...
s 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 \ell_1 + \ell_2 - m.


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 distinct nuclear species (or ''nuclides'') of the same chemical element. They have the same atomic number (number of protons in their Atomic nucleus, nuclei) and position in the periodic table (and hence belong to the same chemica ...
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 commutative product opera ...
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 viewed ...
. Like the cup product, the intersection product is
graded-commutative In Abstract algebra, 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'', ...
.


Examples of transverse intersections

The simplest non-trivial example of transversality is of arcs in a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is not a tangency, i.e., their tangent lines inside the tangent plane to the surface are distinct. In a three-dimensional space, two curves can be transverse only when they have empty intersection, since their tangent spaces could generate at most a two-dimensional space. 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 G is a simple Lie group and \mathfrak is its Lie algebra. By the Jacobson–Morozov theorem every nilpotent element e \in \mathfrak can be included into an \mathfrak-triple (e, h, f) . The representation theory of \mathfrak tells us that \mathfrak = mathfrak, e\oplus \mathfrak_f . The space mathfrak, e is the
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
at e to the adjoint orbit \rm(G)e 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 relat ...
e + \mathfrak_f intersects the orbit of e transversally. The space e + \mathfrak_f is known as the "Slodowy slice" after Peter Slodowy.


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 Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
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 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.


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 of an oriented manifold's
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
—i.e. a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
—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 eve ...
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 o ...
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 (, ) 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 vectors and ...
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 vers ...
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 (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
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 (, ) 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 vectors and ...
!)


Grammar

"Transversal" is a noun; the adjective is "transverse."
quote from J.H.C. Whitehead, 1959Hirsch (1976), p.66


See also

* Transversality theorem


Notes


References

* * * {{DEFAULTSORT:Transversality (Mathematics) Differential topology Calculus of variations Geometry