nLab infinite-dimensional manifold

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:

• Hilbert manifold

• Banach manifold

• ILH manifold

• Fréchet manifold

• convenient manifold

Classes of examples

• manifold structure of mapping spaces

• infinite-dimensional Lie group

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.

Related concepts

• nonlinear functional

• orthogonal structure

• integration over infinite-dimensional manifolds

• pro-manifold

References

General

Textbook accounts:

• Alexander Schmeding, An introduction to infinite-dimensional differential geometry, Cambridge University Press (arXiv:2112.08114)

in terms of Frölicher spaces and diffeological spaces:

• Andreas Kriegl, Peter Michor: The convenient setting of global analysis, AMS (1997)

Aspects of the homotopy theory of infinite-dimensional manifolds:

• Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

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)