nLab infinite-dimensional manifold

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)

Manifolds and cobordisms

Contents

Idea

The basic definition of a manifold (especially a smooth manifold) is as a space locally modeled on a finite-dimensional Cartesian space. This can be generalized to a notion of smooth manifolds locally modeled on infinite-dimensional topological vector spaces. Typical examples of these are mapping spaces between finite-dimensional manifolds, such as loop spaces.

Definitions

See specific versions:

Classes of examples

Properties: Embedding into convenient toposes

Various types of smooth manifolds embed into the quasi-toposes of diffeological spaces and hence the topos of smooth spaces. See there for more.

References

General

Textbook accounts:

in terms of Frölicher spaces and diffeological spaces:

Aspects of the homotopy theory of infinite-dimensional manifolds:

Integration

On integration over infinite-dimensional manifolds (for instance path integrals):

  • Irving Segal, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (Euclid)

  • Hui-Hsiung Kuo, Integration theory on infinite-dimensional manifolds, Transactions of the American Mathematical Society Vol. 159, (Sep., 1971), pp. 57-78 (JSTOR)

  • David Shale, Invariant integration over the infinite dimensional orthogonal group and related spaces, Transactions of the American Mathematical Society Vol. 124, No. 1 (Jul., 1966), pp. 148-157 (JSTOR)

Last revised on December 27, 2021 at 22:30:46. See the history of this page for a list of all contributions to it.