nLab
infinite-dimensional manifold

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

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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

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 August 29, 2016 03:49:13 by Urs Schreiber (89.15.236.152)