×

Differential geometry of frame bundles. (English) Zbl 0673.53001

Let M be an n-dimensional manifold, then \(J^ 1_ nM\) denotes the manifold of all 1-jets at 0 of smooth mappings \({\mathbb{R}}^ n\to M\) and the linear frame bundle FM consists of the open subbundle of all invertible jets. Then there is a natural smooth mapping \(j_ M: J^ 1_ nFM\to FJ^ 1_ nM\) (theorem 2.2.3) which is the key to most of the developments in this book. With its help one can prolong G-structures from the manifold M to a \(J^ 1_ nG\)-structure on the frame bundle FM, check that integrability of one of these structures implies integrability of the other.
Then lifts of linear connections on M to linear connections on FM are studied: treated are the diagonal and horizontal ones. Next the diagonal lift of Riemannian metrics on M to Riemannian metrics on FM is studied. It should be noted, however, that on these topics classifications of all natural lifts are available in the papers at the end of the review which are not cited.
The next chapter is devoted to several types of f-structures on FM. Then follows a treatment of systems of connections in the sense of Mangiarotti and Modugno. This is an axiomatic setting of some of the most relevant properties of the space of all principal connections on a principal bundle which recovers some of the most important results of it: (Generalized) connections on some fibre bundle are just projections onto the vertical bundle. Those in the system are parametrized by sections of a suitable bundle, on which there exists a universal connection and curvature, which generalizes properties of the Liouville form and symplectic form on the cotangent bundle. For principal bundles this universal connection and curvature is due to Garcia. The last chapter is devoted to a repetition of parts of the treatment so far where the functor \(J^ 2_ p=J^ 2_ 0({\mathbb{R}}^ p, )\) replaces \(J^ 1_ n.\)
The book is carefully written, with lots of details and good explanations. But, as the last chapter makes already clear, the setup is too special: most of the development goes through if one replaces \(J^ 1_ n\) by an arbitrary Weil functor (see some of the references below) with easier and clearer proofs. Also the authors just discuss some of the prolongations of structure and ignore the question for a classification of all natural ones.
[References: I. Kolař, J. Natl. Acad. Math. India 5, 127-141 (1987); I. Kolař, Ann. Global Anal. Geom. 6, 109-117 (1988); O. Kowalski, and M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to metrics on linear frame bundles - classification, Differential geometry and its applications, Proc. Conf., Brno/Czech. 1986, Math. Appl., East Eur. Ser. 27, 149-178 (1987; Zbl 0632.53040); M. Sekizawa, Monatsh. Math. 105, No.3, 229-243 (1988; Zbl 0639.53022); J. Slovák, Abstract analysis, Proc. 14th Winter Sch., Srni/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 143-155 (1987; Zbl 0644.53027).]
Reviewer: P.Michor

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C05 Connections (general theory)
53C10 \(G\)-structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A20 Jets in global analysis
PDFBibTeX XMLCite