Fredholm Quadratic Sections

A Fredholm quadratic section on a Hilbert manifold M is a smooth section of the Banach bundle of symmetric bilinear forms whose associated operators a re Fredholm; such a section defines a singular pseudo-Riemannian structure on M. These structures appear naturally after induction on submanifolds of finit codimension in a pseudo-Riemannian Hilbert manifold (W, Q). The germs of singular pseudo-Riemannian Hilbert structures are classified genericall y allowing us to study the impact of the appearance of their locus of degen eracy and furthermore the behavior of the gradient vector fields defined ev erywhere on M. In particular, we describe-the properties of the ideal of sm ooth functions that vanish on this locus and hence we are able to specify a nd construct the extensions of the gradient vector fields on the locus of d egeneracy as well as on the ambient pseudoriemannian hilbert manifold (W, Q ). The morphism of quadratic duality between vector fields and Pfaffian for ms, defines on the tangent bundle TM a singular symplectic structure by pul l-back of the canonical symplectic structure of the cotangent bundle T*M. T he locus of degeneracy of this structure is stratified in Banach submanifol ds and its projection gives exactly the locus of degeneracy of the singular pseudo-Riemannian structure on M. We study the obstructions and the influe nce of its appearance on the comportement of the geodesic spray associated to the induced quadratic form on M. Finally, we signify certain consequence s concerning the associated singular connections and also certain applicati ons related to these structures. This article continues the study of the Fr edholm Symplectic Structures that appeared in the "Bulletin des Sciences Ma thematiques", cf. [An, Pe, Pn]. (C) Elsevier, Paris.