Conjugate Connections and Differential Equations on Infinite Dimensional Manifolds

Abstract

Abstract. On a smooth manifold M, the vector bundle structures of the second order tangent bundle, T 2 M bijectively correspond to linear connections. In this paper we classify such structures for those Fréchet manifolds which can be considered as projective limits of Banach manifolds. We investigate also the relation between ordinary differential equations on Fréchet spaces and the linear connections on their trivial bundle; the methodology extends to solve differential equations on those Fréchet manifolds which are obtained as projective limits of Banach manifolds. Such equations arise in theoretical physics.

Second order structures for sprays and connections on Frechet manifolds

“…Here we try to generalize this to the case of Fréchet manifolds where difficulties arise due to intrinsic problems of the model spaces of these manifolds and mainly due to the inability to solve general differential equations (see [3], [17] and [27]). We show that if one focuses on the category of projective limit manifolds, then similar results can be obtained.…”

“…The study of infinite dimensional manifolds has received much interest due to its interaction with bundle structures, fibrations and foliations, jet fields, connections, sprays, Lagrangians and Finsler structures ( [1], [14], [7], [8], [10], [18] and [30]). In particular, non-Banach locally convex modelled manifolds have been studied from different points of view (see for example [2], [4], [11], [12], [19] and [27]). Fréchet spaces of sections arise naturally as configurations of a physical field and the moduli space of inequivalent configurations of a physical field is the quotient of the infinite-dimensional configuration space X by the appropriate symmetry gauge group.…”

On base conformal warped product manifolds with conjugate connections

“…Also, in [4], the geometry of product conjugate connections has been studied. Furthermore there are many studies deal with this subject as [1], [3], [5]. On the other hand, base conformal warped product manifolds have been studied in [11].…”

Strong projective limit of Banach Lie algebroids

“…These problems have a solution on certain projective limits of spaces: on one hand, existence of integral curves of vector fields, autoparallel curves with respect to linear connections (cf. [4]), horizontal global section for connection on particular spaces (cf. [3]); on the other hand, existence of a generalized Lie group H 0 ðFÞ as structural group for the tangent bundle (cf.…”

“…[4]), horizontal global section for connection on particular spaces (cf. [3]); on the other hand, existence of a generalized Lie group H 0 ðFÞ as structural group for the tangent bundle (cf. [14]).…”