nLab
infinite-dimensional manifold

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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

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)

Revised on October 27, 2013 17:27:50 by Toby Bartels (98.19.40.208)