nLab
Fréchet manifold

Idea

A kind of infinite-dimensional manifold. In analogy to how a smooth manifold is a manifold modeled on a Cartesian space ${ℝ}^n$ in CartSp, a Fréchet manifold is a manifold modeled on a Fréchet space.

Definition

Fréchet manifolds

It is possible to define, analog to the finite dimensional case, the notion of smooth functions of Fréchet spaces, see there. Therefore, the usual definition of a manifold carries over word by word:

Definition

A Fréchet manifold is a Hausdorff topological space with an atlas of coordinate charts taking their value in Fréchet spaces, such that the coordinate transition functions are all smooth functions between Fréchet spaces.

It is possible to generalize some concepts of differential geometry from the finite case to the Fréchet case, one has to be careful, however:

1. The dual of a Fréchet space that is not a Banach space is never a Fréchet space, therefore one cannot e.g. define both the tangent and the cotangent bundle as Fréchet manifolds. More serious is however

2. The existence and uniqueness theorems for ordinary differential equations fail in infinite dimensions, so that theorems depending on that from finite dimensional differential geometry cannot be transscribed to the infinite situation in general. It is possible to do this on a case by case basis however.

Tangent Vectors

There are several definitions of tangent vectors that are equivalent in the finite dimensional setting, but may be different in infinite dimensions. Tangent vectors can be defined to be derivations on germs of functions (algebraic definition), or as equivalence classes of smooth curves (kinematic definition). For the time being we settle with the kinematic definition:

Definition

kinematic tangent vector

The kinematic tangent vector space of a Fréchet manifold $M$ at a point $p$ consists of all pairs $(p,c\prime (0))$ where $c$ is a smooth curve

$$c:ℝ\to M\text{with}c(0)=p$$c: \mathbb{R} \to M \; \text{with} \; c(0) = p

As usual, the set of pairs $(p,c\prime (0)),p\in M$ forms a Fréchet manifold, the tangent bundle $\mathrm{TM}$.

The last sentence makes use of the notion of vector bundle, which can be defined exactly as in the finite dimensional setting:

Vector bundles

Definition

vector bundle

A Fréchet manifold $V$ is a Fréchet vector bundle over $M$ with projection $\pi$, if for every point $p\in M$ there are charts of $M$ and $V$ such that $V$ is mapped locally to $U\subset F\times G$ for Fréchet spaces $F,G$, the projection $\pi$ corresponds to the projection of $U\times G$ to $U$, and the vector space structure on each fibre is that induces by the vector space structure on $G$.

Since, as mentioned before, the dual space of a Fréchet space that is not a Banach space is itself not a Fréchet space, we cannot define the cotangent space canonically as the dual space of the tangent space. Instead we define it directly:

Differential forms

Definition

differential form

A differential form (a one form) $\alpha$ is a smooth map

$$\alpha :TM\to ℝ$$\alpha: T M \to \mathbb{R}

where $\mathrm{TM}$ is the tangent bundle.

Properties

Embedding of Fréchet manifolds into diffeological spaces

We discuss how Fréchet manifolds form a full subcategory of that of diffeological spaces.

Definition

Define a functor

$$\iota :\mathrm{FrechetManifolds}\to \mathrm{DiffeologicalSpaces}$$\iota \colon FrechetManifolds \to DiffeologicalSpaces

from Fréchet manifolds to diffeological spaces in the evident way by taking for $X$ a Fréchet manifold for any $U\in$ CartSp the set of $U$-plots of $\iota (X)$ to be the set of smooth functions $U\to X$.

Proposition

The functor $\iota :\mathrm{FrechetManifolds}\hookrightarrow \mathrm{DiffeologicalSpaces}$ is a full and faithful functor.

This appears as (Losik, theorem 3.1.1).

Proposition

Let $X,Y\in \mathrm{SMoothManifold}$ with $X$ a compact manifold.

Then under this embedding, the diffeological mapping space structure $C^{\mathrm{\infty }}(X,Y{)}_{\mathrm{diff}}$ on the mapping space coincides with the Fréchet manifold structure $C^{\mathrm{\infty }}(X,Y{)}_{\mathrm{Fr}}$:

$$\iota (C^{\mathrm{\infty }}(X,Y{)}_{\mathrm{Fr}})\simeq C^{\mathrm{\infty }}(X,Y{)}_{\mathrm{diff}}.$$\iota(C^\infty(X,Y)_{Fr}) \simeq C^\infty(X,Y)_{diff} \,.

This appears as (Waldorf, lemma A.1.7).

Examples

Standard examples of Fréchet manifolds are smooth mapping spaces such as the

• smooth loop space.

Related concepts

• Banach manifold

• Hilbert manifold, IHL manifold

• convenient manifold

• F-star-algebra

References

For instance

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

• V.I. Arnold, ; B.A. Khesin: Topological methods in hydrodynamics. (Springer 1998, ZMATH)

• Boris Khesin, Robert Wendt: The geometry of infinite-dimensional groups. (Springer 2009, ZMATH)

The embedding into diffeological spaces is due to

• M. V. Losik, Fréchet manifolds as diffeological spaces, Soviet. Math. 5 (1992)

and reviewed in section 3 of

• M. V. Losik, Categorical Differential Geometry Cah. Topol. Géom. Différ. Catég., 35(4):274–290, 1994. (pdf)

The preservation of mapping spaces under this embedding is due to

• Konrad Waldorf Transgression to Loop Spaces and its Inverse I (arXiv:0911.3212)