Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces.

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Some recent work in Frechet geometry

Ambrose, Palais and Singer [6] introduced the concept of second order structures on nite dimensional manifolds. Kumar and Viswanath [23] extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Fr echet manifolds. It is proved that the existence of Christoel and Hessian structures, connections, sprays and dissections are equivalent on those Fr echet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary dierential equations on non-Banach innite dimensional manifolds. Concrete examples of the structures are provided using direct and at connections.

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Second order structures for sprays and connections on Frechet manifolds

We endow projective (resp. direct) limits of Banach tensor structures with Frechet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.

Projective and Direct limits of Banach $G$ and tensor structures

We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group GL(∞,R) = lim→ GL(R n ).

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Geometric structure for the tangent bundles of direct limit manifolds

We define the notion of projective limit of local shift morphisms of type $\left( r,s\right) $ and endow the space of such mathematical objects with an adapted differential structure. The notion of shift Poisson tensor $P$ on a Hilbert tower corresponds to such morphisms which are antisymmetric and whose Schouten bracket $\left[ P,P\right] $ vanishes. We illustrate this notion with the example of the famous KdV equation on the circle $\mathbb{S}^{1}$ for which one can associate a couple of compatible Poisson tensors of this type on the Hilbert tower $\left( H^{n}(\mathbb{S}^{1})\right) _{n\in\mathbb{N}% ^{\ast}}$.

G. N. Galanis

Papers

Conjugate connections and dierential equations on innite dimensional manifolds