nLab smooth map

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A function which is differentiable function to arbitrary order is called a smooth function.

In the real numbers

Epsilon-delta definition

Let \mathbb{R} be the real numbers. A function f:f:\mathbb{R} \to \mathbb{R} is smooth if it comes with a sequence of functions D ()f:()D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R}) and a sequence of functions M ()f:( + +)M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+) in the positive rational numbers, such that

  • for every real number xx \in \mathbb{R}, (D 0f)(x)=f(x)(D^{0}f)(x) = f(x)

  • for every natural number nn \in \mathbb{N}, for every positive rational number ϵ +\epsilon \in \mathbb{Q}_+, for every real number hh \in \mathbb{R} such that 0<|h|<M nf(ϵ)0 \lt | h | \lt M^{n}f(\epsilon), and for every real number xx \in \mathbb{R},

    |(D nf)(x+h)(D nf)(x)h(D n+1f)(x)|<ϵ|h||(D^{n}f)(x + h) - (D^{n}f)(x) - h (D^{n + 1}f)(x)| \lt \epsilon |h|

Unwrapping the recursive definition above, a function f:f:\mathbb{R} \to \mathbb{R} is smooth if it comes with a sequence of functions D ()f:()D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R}) and a sequence of functions M ()f:( + +)M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+) in the positive rational numbers, such that

  • for every real number xx \in \mathbb{R}, (D 0f)(x)=f(x)(D^{0}f)(x) = f(x)

  • for every natural number nn \in \mathbb{N}, for every positive rational number ϵ +\epsilon \in \mathbb{Q}_+, for every real number hh \in \mathbb{R} such that 0<|h|<M nf(ϵ)0 \lt | h | \lt M^{n}f(\epsilon), and for every real number xx \in \mathbb{R},

    |f(x+h) i=0 nh i(D if)(x)i!|<ϵ|h n|\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|

 Infinitesimal definition

Given a predicate PP on the real numbers \mathbb{R}, let II denote the set of all elements in \mathbb{R} for which PP holds. A partial function f:f:\mathbb{R} \to \mathbb{R} is equivalently a function f:If:I \to \mathbb{R} for any such predicate PP and set II.

A function f:If:I \to \mathbb{R} is smooth at a subset SIS \subseteq I with injection j:Sj:S \hookrightarrow \mathbb{R} if it has a function d fdx :×S\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R} with d 0fdx 0(a)=a\frac{d^0 f}{d x^0}\left(a\right) = a for all aSa \in S, such that for all Archimedean ordered Artinian local \mathbb{R}-algebras AA with ring homomorphism h A:Ah_A:\mathbb{R} \to A and nilradical DD, natural numbers nn \in \mathbb{N}, and purely infinitesimal elements ϵD\epsilon \in D such that ϵ n+1=0\epsilon^{n + 1} = 0

f A(h A(j(a))+ϵ)= i=0 n1i!h A(d ifdx i(a))ϵ if_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i

Equivalently, let [[ϵ]]\mathbb{R}[[\epsilon]] denote the ring of univariate formal power series on \mathbb{R}. [[ϵ]]\mathbb{R}[[\epsilon]] is a Artinian local \mathbb{R}-algebra with homomorphism h:[[ϵ]]h:\mathbb{R} \to \mathbb{R}[[\epsilon]]. A function f:If:I \to \mathbb{R} is smooth at a subset SIS \subseteq I with injection j:Sj:S \hookrightarrow \mathbb{R} if it has a function d fdx :×S\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R} with d 0fdx 0(a)=a\frac{d^0 f}{d x^0}\left(a\right) = a for all aSa \in S, such that for all natural numbers nn \in \mathbb{N}

f A(h(j(a))+ϵ)= i=0 1i!h(d ifdx i(a))ϵ if_A(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i

A function f:If:I \to \mathbb{R} is smooth at an element aIa \in I if it is smooth at the singleton subset {a}\{a\}, and a function f:If:I \to \mathbb{R} is smooth if it is smooth at the improper subset of II.

Between Cartesian spaces

A function on (some open subset of) a cartesian space n\mathbb{R}^n with values in the real line \mathbb{R} is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if A nA \subseteq \mathbb{R}^n is any subset, a function f:Af: A \to \mathbb{R} is defined to be smooth if it has a smooth extension to an open subset containing AA.

By coinduction: A function f:f : \mathbb{R} \to \mathbb{R} is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

For A nA \subseteq \mathbb{R}^n, a smooth map ϕ:A m\phi: A \to \mathbb{R}^m is a function such that πϕ\pi \circ \phi is a smooth function for every linear functional π: m\pi: \mathbb{R}^m \to \mathbb{R}. (In the case of finite-dimensional codomains as here, it suffices to take the π\pi to range over the mm coordinate projections.)

The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).

Between smooth manifolds

A topological manifold whose transition functions are smooth maps is a smooth manifold. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of atlases, hence which is a kk-fold differentiable function (see there for more details), for all kk The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps between them.

Between generalized smooth spaces

There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.

For details see for example at smooth set.

Properties

Basic facts about smooth functions are

Examples

Every analytic functions (for instance a holomorphic function) is also a smooth function.

A crucial property of smooth functions, however, is that they contain also bump functions.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:

  • Lev Pontrjagin, Chapter I of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

Last revised on October 5, 2023 at 04:38:52. See the history of this page for a list of all contributions to it.