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Quasi-tangent \(p\)-structures and quasi-cotangent \(p\)-structures. (\(p\)-estructuras casi tangentes y \(p\)-estructuras casi cotangentes.) (Spanish) Zbl 0839.53022

Publicaciones del Departamento de Geometría y Topología, Universidad de Santiago de Compostela. 78. Santiago de Compostela: Univ., Dept. de Geometría y Topología. 166 p. (1990).
This monograph contains the doctoral dissertation of the author. The purpose is to generalize two classes of important geometrical structures: almost tangent structures and almost cotangent structures. The first were obtained by abstracting the geometry of a tangent bundle, which is given by the vertical endomorphism, and the latter by abstracting the geometrical ingredients of a cotangent bundle, namely its canonical symplectic structure and the vertical bundle (in other words, its canonical polarization). \(p\)-almost tangent and cotangent structures are useful in the geometric formulation of classical field theory [see for instance the reviewer and P. R. Rodrigues, Lett. Math. Phys. 14, 353-362 (1987; Zbl 0637.53056); J. Math. Phys. 30, No. 6, 1351-1353 (1989; Zbl 0689.53018)and the references of Awane and Norris quoted below].
The monograph is divided into five chapters. In Chapter 1, \(p\)-almost tangent structures are introduced. The geometrical model is the so-called tangent bundle of \(p^1\)-velocities, \(T^1_p M\), of a manifold \(M\), introduced by Ehresmann. There exists a set of vertical endomorphisms \(\{J_1, \dots, J_p\}\) on \(T^1_p M\) which generalize the vertical endomorphism \(J\) on \(TM\). Moreover, \(T^1_p M\) admits several fibrations over \(T^1_{p-1} M\) and \(M\). A \(p\)-almost tangent structure is defined as a family of \(p\) \((1,1)\)-tensor fields \(\{J_1, \dots, J_p\}\) on a manifold \(N\) satisfying some compatible relations. Such a structure is identified as a \(G\)-structure and its integrability is studied. A large family of (compact and non-compact) examples is given. The group of automorphisms of a \(p\)-almost tangent structure is proved to be a Lie group.
Chapter 2 is devoted to study a particular kind of \(p\)-almost tangent structures, those which define a fibration. This is, the corresponding \(G\)-structure is integrable and the foliation induced by \(V= \text{Im } J_1 \oplus \dots \oplus \text{Im } J_p\) defines a fibration \(N\to M= N/V\). It is proved that, under certain global hypotheses, \(N\) is isomorphic to the tangent bundle of \(p^1\)-velocities of \(M\). In Chapter 3, the author establishes some obstruction results to the existence of \(p\)-almost tangent structures by using the theory of characteristic classes.
Chapters 4 and 5 are devoted to study the dual version of \(p\)-almost tangent structures, the so-called \(p\)-almost cotangent structures. The geometrical model is the tangent bundle of \(p^1\)-covelocities, \(T^*_{p^1} M\), of a manifold \(M\), also introduced by Ehresmann. There exists a family of presymplectic 2-forms \(\{\omega_1, \dots, \omega_p\}\) on \(T^*_{p^1} M\). If \(p=1\), \(\omega_1\) is just the canonical symplectic form of the cotangent bundle \(T^* M\). Again, \(T^*_{p^1} M\) admits several fibrations which define a family of distributions \(\{V_1, \dots, V_p\}\). For \(T^* M\), they reduce to the vertical distribution. Thus, a \(p\)-almost cotangent structure on a manifold \(N\) consists of a family of \(p\) presymplectic forms \(\{\omega_1, \dots, \omega_p\}\) and a family of \(p\)-distributions \(\{V_1, \dots, V_p\}\) on \(N\) satisfying some compatibility relations. Such a structure is identified as a \(G\)-structure and its integrability is studied. A lot of compact and non-compact examples is given.
In Chapter 5, a particular kind of \(p\)-almost cotangent structures is studied, those called regular. They define fibrations over the space of leaves \(M= N/V\), where \(V= V_1 \oplus\dots \oplus V_p\). Under certain global hypotheses, the author proves that \(N\) is isomorphic to \(T^*_{p^1} M\). An alternative characterization was recently given in [the reviewer, E. Merino, J. A. Oubiña and M. Salgado, Ann. Inst. Henri Poincaré, Phys. Théor. 61, No. 1, 1-15 (1994; Zbl 0815.53039)].
Most of the results contained in her monograph were independently published: the reviewer, I. Méndez and M. Salgado [Rend. Circ. Mat. Palermo, II. Ser. 37, No. 2, 282-294 (1988; Zbl 0672.53040); J. Korean Math. Soc. 25, No. 2, 273-287 (1988; Zbl 0657.53021); Acta Math. Hung. 58, No. 1/2, 45-54 (1991; Zbl 0757.53015); Boll. Unione Mat. Ital., VII. Ser. A 7, No. 1, 97-107 (1993; Zbl 0777.53033)], L. A. Cordero, the reviewer, I. Méndez and M. Salgado [Czech. Math. J. 42, No. 2, 225-234 (1992; Zbl 0811.53026)].
It should be noticed that integrable \(p\)-almost cotangent structures were independently studied by A. Awane [J. Geom. Phys. 13, No. 2, 139-157 (1994; Zbl 0795.58015)]who called them \(p\)-symplectic structures. Recently, L. K. Norris discovered it [Generalized symplectic geometry on the frame bundle of a manifold, Proc. Symp. Pure Math. 54, Part 2, 435-465 (1993; Zbl 0790.53033)]. In fact, the structure defined by Norris is just the induced \(n\)-almost cotangent structure on the bundle of linear coframes \(F^* M\) of an \(n\)-dimensional manifold \(M\), taking into account that \(F^* M\) is an open and dense subset of \(T^*_{n^1} M\). Since \(F^* M\) and the linear frame bundle \(FM\) are canonically isomorphic, an \(n\)-almost cotangent structure is canonically defined on \(FM\).

MSC:

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