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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.

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.

Reviewer: Marian Ioan Munteanu (Roma)

### MSC:

58B25 | Group structures and generalizations on infinite-dimensional manifolds |

58A20 | Jets in global analysis |

53C05 | Connections (general theory) |

### Keywords:

infinite-dimensional manifolds; Banach manifolds; Fréchet manifolds; second order tangent bundle### Citations:

Zbl 0508.53041
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XMLCite

\textit{C. T. J. Dodson} and \textit{G. N. Galanis}, J. Geom. Phys. 52, No. 2, 127--136 (2004; Zbl 1076.58002)

### References:

[1] | Dodson, C. T.J.; Radivoiovici, M. S., Tangent and frame bundles of order two, Analele Stiintifice ale Universitatii Al. I. Cuza, 28, 63-71 (1982) · Zbl 0508.53041 |

[2] | Galanis, G., Projective limits of vector bundles, Portugaliae Math., 55, 11-24 (1998) · Zbl 0904.58002 |

[3] | Galanis, G., Projective limits of Banach-Lie groups, Periodica Math. Hungarica, 32, 179-191 (1996) · Zbl 0866.58009 |

[4] | A. Kriegl, P. Michor, The Convenient Setting of Global Analysis, vol. 53, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.; A. Kriegl, P. Michor, The Convenient Setting of Global Analysis, vol. 53, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. · Zbl 0889.58001 |

[5] | Leslie, J. A., On a differential structure for the group of diffeomorphisms, Topology, 46, 263-271 (1967) · Zbl 0147.23601 |

[6] | Leslie, J. A., Some frobenius theorems in global analysis, J. Diff. Geom., 42, 279-297 (1968) · Zbl 0169.53201 |

[7] | Vassiliou, E.; Galanis, G., A generalized frame bundle for certain Fréchet vector bundles and linear connections, Tokyo J. Math., 20, 129-137 (1997) · Zbl 0894.58006 |

[8] | Vilms, J., Connections on tangent bundles, J. Diff. Geom., 41, 235-243 (1967) · Zbl 0162.53603 |

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