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Zbl 1121.58007
Galanis, George N.
Differential and geometric structure for the tangent bundle of a projective limit manifold. (English)
[J] Rend. Semin. Mat. Univ. Padova 112, 103-115 (2004). ISSN 0041-8994

The paper under review is within the framework of Fréchet modelled infinite dimensional manifolds, in short Fréchet manifolds, where problems appear in an intrinsic way: the solvability of even linear differential equations may not exist or there may be more than one solution for the same initial values, for instance. The author selects a subclass of Fréchet manifolds -- projective limits of Banach manifolds -- to prove that a vector bundle structure can be defined on its tangent manifold. He shows than on such manifolds connections, generalized Christoffel symbols with values in the Fréchet base space, parallel translation along curves on this manifold with respect to the connection and a corresponding holonomy group can be defined. Compare {\it M.C. Abbati} and {\it A. Maniá} [On differential structure for projective limits of manifolds. J. Geom. Phys. 29, No. 1-2, 35--63 (1999; Zbl 0935.58008)]. Galanis' constructions keep all fundamental geometrical characteristics.
[L. Del Riego (San Luis Potos\'\i)]

MSC 2000:: *58B20 Geometric structures on infinite-dimensional manifolds
53C07 Special connections and metrics on vector bundles
53C15 Geometric structures on manifolds
53C05 Connections, general theory

Citations: Zbl 0935.58008

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