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Second order tangent bundles of infinite dimensional manifolds. (English) Zbl 1076.58002

The second order tangent bundle \(T^2M\) of a smooth manifold \(M\) consists of the equivalent classes of curves on \(M\) that agree up to their acceleration. It is known [C. T. J. Dodson and M. S. Radivoiovici, An. Stiint. Univ. Al. I. Cuza Iasi, N. Ser., Sect. Ia 28, 63–71 (1982; Zbl 0508.53041)] that in the case of a finite \(n\)-dimensional manifold \(M\), \(T^2M\) becomes a vector bundle over \(M\) if and only if \(M\) is endowed with a linear connection. More precisely, \(T^2M\) becomes then and only then a vector bundle over \(M\) with the structure group \(\text{GL}(2n;R)\) and, therefore, a \(3n\)-dimensional manifold.
In this paper the authors extend the results to a wide class of infinite dimensional manifolds. First, it is considered a manifold \(M\) modeled on an arbitrary chosen Banach space \(E\) and it is proven that \(T^2M\) can be thought as a Banach vector bundle over \(M\) with the structure group \(GL(E\times E)\) if and only if \(M\) admits a linear connection. Then the authors study the case of Fréchet (non-Banach) modeled manifolds. In this framework things proved much more complicated. However, by restriction to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds, it is possible to endow \(T^2M\) with a vector bundle structure over \(M\) with a new topological groups as structure group which replace the pathological general linear group of the fibre type. This construction is equivalent with the existence of a specific type of linear connection on \(M\) characterized by a generalized set of Christoffel symbols.

MSC:

58B25 Group structures and generalizations on infinite-dimensional manifolds
58A20 Jets in global analysis
53C05 Connections (general theory)

Citations:

Zbl 0508.53041
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References:

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