This article needs additional citations for verification .(December 2009) |
In mathematics, 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. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.
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. [1] 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.
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
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. [2] If the maps are embeddings, this is equivalent to transversality of submanifolds.
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:
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 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 on cohomology. Like the cup product, the intersection product is graded-commutative.
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, 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 is a simple Lie group and is its Lie algebra. By the Jacobson–Morozov theorem every nilpotent element can be included into an -triple . The representation theory of tells us that . The space is the tangent space at to the adjoint orbit and so the affine space intersects the orbit of transversally. The space is known as the "Slodowy slice" after Peter Slodowy.
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:
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.
Using Sard's theorem, 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—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 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 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 bundle over the space of maps from a Riemann surface 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 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!)
"Transversal" is a noun; the adjective is "transverse."
quote from J.H.C. Whitehead, 1959 [3]
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line".
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.
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 as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar and a vector, a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R3, the Gauss map is a map N: X → S2 (where S2 is the unit sphere) such that for each p in X, the function value N(p) is a unit vector orthogonal to X at p. The Gauss map is named after Carl F. Gauss.
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.
In Riemannian geometry, the geodesic curvature of a curve measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold , the geodesic curvature is just the usual curvature of . However, when the curve is restricted to lie on a submanifold of , geodesic curvature refers to the curvature of in and it is different in general from the curvature of in the ambient manifold . The (ambient) curvature of depends on two factors: the curvature of the submanifold in the direction of , which depends only on the direction of the curve, and the curvature of seen in , which is a second order quantity. The relation between these is . In particular geodesics on have zero geodesic curvature, so that , which explains why they appear to be curved in ambient space whenever the submanifold is.
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of at to the tangent space of at , . Hence it can be used to push tangent vectors on forward to tangent vectors on . The differential of a map is also called, by various authors, the derivative or total derivative of .
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion.
In mathematics, a differentiable 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 then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.
In gravitation theory, a world manifold endowed with some Lorentzian pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over a world manifold.
In differential geometry, a Lie group action is a group action adapted to the smooth setting: is a Lie group, is a smooth manifold, and the action map is differentiable.