Manifolds over function algebras and mapping spaces (Q750965)

The author proposes a new category for dealing with mapping spaces. The space of smooth maps f: \(M\to N\) from a compact manifold to another manifold is an infinite-dimensional manifold modelled on the space \(C^{\infty}(M,f^*TN)\) of smooth sections in the pullback of the tangent bundle. The space \(C^{\infty}(M,f^*TN)\) is a Fréchet space, but the general theory of Fréchet spaces is not very effective, so normally one completes the space to a Banach or a Hilbert space. There have been several attempts to construct a general theory to deal with the spaces of \(C^{\infty}\)-mappings, e.g., the category of ILH-manifolds [cf. \textit{H. Omori}, Infinite-dimensional Lie transformation groups, Lect. Notes Math. 427 (1974; Zbl 0328.58005)] or the Nash-Moser category [cf. \textit{R. S. Hamilton}, Bull. Am. Math. Soc., New Ser. 7, 65-222 (1982; Zbl 0499.58003)]. The author defines the category of A-manifolds, and develops the basic theory. An A-manifold is modelled on the space \(C^{\infty}(M,V)\) of smooth sections in a vector bundle, but an A-map \(C^{\infty}(M,V)\to C^{\infty}(M,V')\) is not just any smooth map, but shall be induced by a map \(V\to V'\) which covers the identity. As a consequence there is no partition of unity in the theory and the objects one constructs on \(C^{\infty}(M,N)\) are induced by objects on M and N. For example, if M has a volume element and N a Riemannian metric, then it is possible to define a Riemannian A-metric on \(C^{\infty}(M,N)\). One gets the usual real Riemannian metric by integrating the A-metric, and the author demonstrates how it is possible to compute the curvature without completing the spaces to Banach manifolds. Some objects involving derivatives fall outside the category of A-manifolds, but then it is possible to define the larger category of \(A^{(r)}\)-manifolds which are modelled on the space of sections in the rth jet bundle.