Connections and differential calculus on fibred manifolds. 3rd ed. of the preprint.

*(English)*Zbl 0841.53023
Florence: Univ., Dept. of Applied Mathematics, 142 p. (1989).

The authors develop the theory of connections on an arbitrary fibred manifold \(E \to B\) by using the Frölicher-Nijenhuis bracket and outline its applications in field theories of mathematical physics. Another organizing idea used in the paper is the concept of system of a fibred manifold, which was introduced by the second author.

In the beginning, the iterated tangent and jet prolongations of \(E\) are studied. In the case of an affine bundle, the authors describe the polynomial maps between the individual fibres. The Frölicher-Nijenhuis bracket of two tangent valued forms on a manifold \(M\) is defined by means of an original formula, which generalizes the classical eight-terms expression of the case of two one-forms. This bracket induces a graded Lie algebra structure on the space of all tangent valued forms. Given a tangent valued \(k\)-form \(\vartheta\), the Nijenhuis differential of tangent valued forms is defined by bracketing with \(\vartheta\). Then the authors deduce that a metric on \(M\) induces the covariant codifferential and the Laplacian analogously to the classical Hodge theory. These constructions are extended to systems of tangent valued forms as well. The most important special case is that of a projectable tangent valued one-form on \(E\) over the identity of \(TB\), which is equivalent to a connection on \(E\). In such a case, the previous calculus leads to deep results about arbitrary connections on \(E\). Special attention is paid to polynomial connections on an affine bundle. The additional structures on the tangent bundle \(TM \to M\) of a manifold are discussed in detail. The possibility of applications of such a geometric theory in mathematical physics is demonstrated on a generalization of the Maxwell field coupled with a current. All geometric problems are discussed in the intrinsic form, but the coordinate expressions are always presented. This makes the paper attractive for both geometers and physicists.

In the beginning, the iterated tangent and jet prolongations of \(E\) are studied. In the case of an affine bundle, the authors describe the polynomial maps between the individual fibres. The Frölicher-Nijenhuis bracket of two tangent valued forms on a manifold \(M\) is defined by means of an original formula, which generalizes the classical eight-terms expression of the case of two one-forms. This bracket induces a graded Lie algebra structure on the space of all tangent valued forms. Given a tangent valued \(k\)-form \(\vartheta\), the Nijenhuis differential of tangent valued forms is defined by bracketing with \(\vartheta\). Then the authors deduce that a metric on \(M\) induces the covariant codifferential and the Laplacian analogously to the classical Hodge theory. These constructions are extended to systems of tangent valued forms as well. The most important special case is that of a projectable tangent valued one-form on \(E\) over the identity of \(TB\), which is equivalent to a connection on \(E\). In such a case, the previous calculus leads to deep results about arbitrary connections on \(E\). Special attention is paid to polynomial connections on an affine bundle. The additional structures on the tangent bundle \(TM \to M\) of a manifold are discussed in detail. The possibility of applications of such a geometric theory in mathematical physics is demonstrated on a generalization of the Maxwell field coupled with a current. All geometric problems are discussed in the intrinsic form, but the coordinate expressions are always presented. This makes the paper attractive for both geometers and physicists.

Reviewer: I.Kolář (Brno)

##### MSC:

53C05 | Connections, general theory |

58A20 | Jets in global analysis |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |