Existence of non–first-order connections in infinite dimensions

Determine whether there exist connections (covariant derivatives) on vector bundles over infinite-dimensional manifolds, modeled on locally convex spaces in the Bastiani calculus setting, that are not first order in the sense that the value at a point depends on more than the 1-jet of the section at that point; equivalently, construct such a connection or prove that all connections on infinite-dimensional manifolds are first order.

Background

In finite dimensions, every connection on a vector bundle is of first order, meaning its value at a point depends only on the value of the vector field and the 1-jet of the section at that point. This property is standard and underpins many constructions in Riemannian geometry and optimization.

In the infinite-dimensional setting considered in this paper, the authors note that the finite-dimensional proof does not generalize without further assumptions. They show that connections induced by metric sprays are first order, but it remains unresolved whether there exist connections that fail this property in general on infinite-dimensional manifolds.

References

It is a standard argument that every connection ∇ on a finite-dimensional vector bundle is of first order in the sense that for section X,Y and m ∈ M, the value ∇_X Y (m) depends only on the value X(m) and the first order jet of Y. Unfortunately, the finite-dimensional proof does not generalise without further assumptions. One can prove that every connection associated to a spray, cf. Cref{App:diffgeo}, is a first order connection in this sense. It is unknown whether there exist connections on infinite-dimenisonal manifolds which are not of first order.

— Optimization on Weak Riemannian Manifolds  (2603.25396 - Zalbertus et al., 26 Mar 2026) in Remark rem:fo-connection, Section 4.2 (Second-Order Optimality Conditions)