maps of manifolds
   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 ...
, more specifically in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
and
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, various types of
maps A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
or functions between
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s are studied, both as objects in their own right and for the light they shed.


Types of maps

Just as there are various types of manifolds, there are various types of maps of manifolds. In
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, the basic types of maps correspond to various categories of manifolds: DIFF for
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...
s between
differentiable 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 ...
s, PL for
piecewise linear function In mathematics, a piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. Definition A piecewise linear function is a function defined on a (possibly unbounded) ...
s between piecewise linear manifolds, and TOP for
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s between
topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...
s. These are progressively weaker structures, properly connected via PDIFF, the category of
piecewise In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...
-smooth maps between piecewise-smooth manifolds. In addition to these general categories of maps, there are maps with special properties; these may or may not form categories, and may or may not be generally discussed categorically. In
geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
a basic type 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, of which
knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
is a central example, and generalizations such as
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, submersions,
covering space In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
s, and ramified covering spaces. Basic results include the
Whitney embedding theorem In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real - dimensional manifold (required also to be Hausdorf ...
and
Whitney immersion theorem In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for m>1, any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space, an ...
. In complex geometry, ramified covering spaces are used to model
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 ...
s, and to analyze maps between surfaces, such as by the Riemann–Hurwitz formula. In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and
Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let ( ...
s; a basic result is the
Nash embedding theorem The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedding, embedded into some Euclidean space. Isometry, Isometric means preserving the length of ever ...
.


Scalar-valued functions

A basic example of maps between manifolds are scalar-valued functions on a manifold, \scriptstyle f\colon M \to \mathbb or \scriptstyle f\colon M \to \mathbb, sometimes called
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a reg ...
s or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold. In geometric topology, most commonly studied are
Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
s, which yield handlebody decompositions, which generalize to Morse–Bott functions and can be used for instance to understand classical groups, such as in Bott periodicity. In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, one often studies solution to
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
, an important example of which is
harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, where one studies
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
s: the kernel of the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...
. This leads to such functions as the
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics co ...
, and to
heat kernel In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum ...
methods of studying manifolds, such as
hearing the shape of a drum In theoretical mathematics, the conceptual problem of "hearing the shape of a drum" refers to the prospect of inferring information about the shape of a hypothetical idealized drumhead from the sound it makes when struck, i.e. from analysis of ...
and some proofs of the Atiyah–Singer index theorem. The
monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
around a singularity or
branch point In the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valu ...
is an important part of analyzing such functions.


Curves and paths

Dual to scalar-valued functions – maps \scriptstyle M \to \mathbb – are maps \scriptstyle \mathbb{R} \to M, which correspond to curves or paths in a manifold. One can also define these where the domain is an interval ,b especially the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
,1 or where the domain is a circle (equivalently, a periodic path) ''S''1, which yields a loop. These are used to define the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
, chains in
homology theory In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
,
geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
curves, and
systolic geometry In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and ...
. Embedded paths and loops lead to
knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
, and related structures such as links,
braids A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strand ...
, and tangles.


Metric spaces

Riemannian manifold 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 N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s are special cases of metric spaces, and thus one has a notion of
Lipschitz continuity In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...
,
Hölder condition In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that , f(x) - f(y) , \leq C\, x - y\, ^ for all and in the do ...
, together with a
coarse structure In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topolo ...
, which leads to notions such as coarse maps and connections with
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...
.


See also

* :Maps of manifolds