Contents

Idea

A Fréchet manifold modeled on $\mathbb{R}^\infty \coloneqq \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$ has the property that a smooth function out of it is locally (on a neighbourhood of each point) given by a function that depends only on a finite number of the coordinate functions on $\mathbb{R}^\infty$ (Michor 80, 9.5.9). Hence locally such an infinite-dimensional manifold looks like a pro-object in the category of finite-dimensional smooth manifolds.

The key examples of locally pro-manifolds are jet bundles of fiber bundles of finite rank (Saunders 89, Michor 80).

References

• Floris Takens, A global version of the inverse problem of the calculus of variations, J. Differential Geom. Volume 14, Number 4 (1979), 543-562. (Euclid)

• Peter Michor, Manifolds of differentiable mappings, Shiva Publishing, Orpington, 1980.

• D. J. Saunders, The Geometry of Jet Bundles, vol. 142 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1989.

• Igor Khavkine, Urs Schreiber, section 2.2 of Synthetic geometry of differential equations: I. Jets and comonad structure (arXiv:1701.06238)