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:
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:
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
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.
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:
kinematic tangent vector
The kinematic tangent vector space of a Fréchet manifold at a point consists of all pairs where is a smooth curve
As usual, the set of pairs forms a Fréchet manifold, the tangent bundle .
The last sentence makes use of the notion of vector bundle, which can be defined exactly as in the finite dimensional setting:
A Fréchet manifold is a Fréchet vector bundle over with projection , if for every point there are charts of and such that is mapped locally to for Fréchet spaces , the projection corresponds to the projection of to , and the vector space structure on each fibre is that induces by the vector space structure on .
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:
A differential form (a one form) is a smooth map
where is the tangent bundle.
Define a functor
The functor is a full and faithful functor.
This appears as (Losik, theorem 3.1.1).
Let with a compact manifold.
Then under this embedding, the diffeological mapping space structure on the mapping space coincides with the Fréchet manifold structure :
This appears as (Waldorf, lemma A.1.7).
Standard examples of Fréchet manifolds are smooth mapping spaces such as the
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
and reviewed in section 3 of
The preservation of mapping spaces under this embedding is due to