##
**Natural operations in differential geometry.**
*(English)*
Zbl 0782.53013

Berlin: Springer-Verlag. vi, 434 p. (corrected electronic version) (1993).

This book is a first treatise on natural bundles and natural operators. Natural bundles are also called bundle functors. The typical problem studied in the book is the following. Given two natural bundles over \(m\)- manifolds, \(F,G: {\mathcal M} f_ m\to {\mathcal F}{\mathcal M}\), where \({\mathcal M} f_ m\) is the category of \(m\)-dimensional manifolds and local diffeomorphisms and \({\mathcal F}{\mathcal M}\) is the category of fibered manifolds and fibered maps, a natural operator \(A: F\to G\) is a system of natural operators \(A_ M: \text{Sect}(FM)\to\text{Sect}(GM)\), \(\text{Sect}(HM)\) being the set of sections of the fibered manifold \(HM\to M\), satisfying some properties of naturality and locality. The question is: how many natural operators are there between \(F\) and \(G\)? The answer is given taking into account the bijective correspondence between natural operators of finite order \(k\) and \(G^ q_ m\)-invariant maps \(T^ k_ m S\to S'\), where \(q=\max\{r+ k,r'\}\), \(r\) being the order of \(F\), \(r'\) the order of \(G\) and \(S\), \(S'\) the standard fibers of \(F\) and \(G\), respectively. The authors illustrate the power of this procedure with many examples. For instance, the exterior derivative \(d: \Lambda^ k T^*\to \Lambda^{k+1} T^*\) is a first order natural operator and the only natural operators of this kind are the constant multiples of \(d\). This is the flavour of the book.

The book is divided in twelve chapters. The first three chapters are devoted to give an introduction to the basic notions in Differential Geometry. However, the content is not completely standard and there are some interesting subjects. In fact, the theory of derivations of Frölicher-Nijenhuis is introduced as well as the so-called Frölicher- Nijenhuis bracket. Also in the third chapter a holonomy theorem for general connections is proved. This theorem generalizes the well-known holonomy theorem of Ambrose-Singer for principal connections. Chapter IV introduces jets on manifolds and an interesting study of jet groups \(G^ k_ m\) is done. The prolongation of principal bundles are defined and the first results on natural bundles and operators are given. The finite order theorems (Peetre theorem and its nonlinear generalizations) are given in Chapter V. In Chapter VI the authors introduce the algebraic and analytical methods in order to find natural operators and some applications are given in Chapter VII. For instance, all bilinear operators of the Frölicher-Nijenhuis type are obtained. Chapter VIII is devoted to study the so-called Weil functors and their natural transformations. Essentially, every product preserving functor is a Weil functor. Then the next chapter IX is devoted to study bundle functors which do not preserve products. Significative differences between both kinds of functors are remarked. In Chapter X the prolongations of vector fields and connections into vector fields and connections are studied. This is a widely studied topic in the literature. The generalized Lie derivative is considered in Chapter XI and some applications to higher order variational calculus are given. Some results on the existence of the Poincaré-Cartan forms were recently obtained by the first author [Natural operators related with the variational calculus, Proc. Conf. Differ. Geom. Appl. Opava (1992)]. Finally, Chapter XII is devoted to study gauge natural bundles and their operators between them. In fact, natural bundles coincide with the associated fiber bundles to higher order frame bundles. Now, we need to consider an arbitrary Lie group and we deduce that gauge natural bundles are associated bundles with the prolongations of principal bundles.

As the authors claimed in the introduction, the book collects many results mainly due to mathematicians of Middle Europe. I agree with this assertion! I recommend the book to those mathematicians and physicists interested in the theory of bundles, connections, differential operators, etc. I only have some minor criticisms. In fact, some references should be added to the bibliography. For instance, the book of S. Kobayashi [Transformation groups in differential geometry (Berlin 1972; Zbl 0246.53031)] contains additional material to Chapter IV. The pioneer paper of P. L. García [Rend. Semin. Mat. Univ. Padova 47, 227- 242 (1972; Zbl 0251.53024)] is a needed reference to universal connections. The diffeomorphism \(TT^*\cong T^* T\) appears for the first time in W. M. Tulczyjew [C. R. Acad. Sci., Paris, Sér. A 283, 675-678 (1976; Zbl 0334.58008)].

The book is divided in twelve chapters. The first three chapters are devoted to give an introduction to the basic notions in Differential Geometry. However, the content is not completely standard and there are some interesting subjects. In fact, the theory of derivations of Frölicher-Nijenhuis is introduced as well as the so-called Frölicher- Nijenhuis bracket. Also in the third chapter a holonomy theorem for general connections is proved. This theorem generalizes the well-known holonomy theorem of Ambrose-Singer for principal connections. Chapter IV introduces jets on manifolds and an interesting study of jet groups \(G^ k_ m\) is done. The prolongation of principal bundles are defined and the first results on natural bundles and operators are given. The finite order theorems (Peetre theorem and its nonlinear generalizations) are given in Chapter V. In Chapter VI the authors introduce the algebraic and analytical methods in order to find natural operators and some applications are given in Chapter VII. For instance, all bilinear operators of the Frölicher-Nijenhuis type are obtained. Chapter VIII is devoted to study the so-called Weil functors and their natural transformations. Essentially, every product preserving functor is a Weil functor. Then the next chapter IX is devoted to study bundle functors which do not preserve products. Significative differences between both kinds of functors are remarked. In Chapter X the prolongations of vector fields and connections into vector fields and connections are studied. This is a widely studied topic in the literature. The generalized Lie derivative is considered in Chapter XI and some applications to higher order variational calculus are given. Some results on the existence of the Poincaré-Cartan forms were recently obtained by the first author [Natural operators related with the variational calculus, Proc. Conf. Differ. Geom. Appl. Opava (1992)]. Finally, Chapter XII is devoted to study gauge natural bundles and their operators between them. In fact, natural bundles coincide with the associated fiber bundles to higher order frame bundles. Now, we need to consider an arbitrary Lie group and we deduce that gauge natural bundles are associated bundles with the prolongations of principal bundles.

As the authors claimed in the introduction, the book collects many results mainly due to mathematicians of Middle Europe. I agree with this assertion! I recommend the book to those mathematicians and physicists interested in the theory of bundles, connections, differential operators, etc. I only have some minor criticisms. In fact, some references should be added to the bibliography. For instance, the book of S. Kobayashi [Transformation groups in differential geometry (Berlin 1972; Zbl 0246.53031)] contains additional material to Chapter IV. The pioneer paper of P. L. García [Rend. Semin. Mat. Univ. Padova 47, 227- 242 (1972; Zbl 0251.53024)] is a needed reference to universal connections. The diffeomorphism \(TT^*\cong T^* T\) appears for the first time in W. M. Tulczyjew [C. R. Acad. Sci., Paris, Sér. A 283, 675-678 (1976; Zbl 0334.58008)].

Reviewer: M.de León (Madrid)

### MSC:

53A55 | Differential invariants (local theory), geometric objects |

58A20 | Jets in global analysis |

53C05 | Connections (general theory) |

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