Construction of the holonomy invariant foliated cocycles on the tangent bundle via formal integrability

Turk J Math (2019) 43: 81 – 102 © TÜBİTAK doi:10.3906/mat-1705-19 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Construction of the holonomy invariant foliated cocycles on the tangent bundle via formal integrability Fatemeh AHANGARI∗, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran Received: 04.05.2017 • Accepted/Published Online: 23.10.2018 • Final Version: 18.01.2019 Abstract: This paper is dedicated to exhaustive structural analysis of the holonomy invariant foliated cocycles on the tangent bundle of an arbitrary (m + n) -dimensional manifold. For this purpose, by applying Spencer theory of formal integrability, sufficient conditions for the metric associated with the semispray S are determined to extend to a transverse metric for the lifted foliated cocycle on T M . Accordingly, this geometric structure converts to a holonomy invariant foliated cocycle on the tangent space, which is totally adapted to the Helmholtz conditions. Key words: Foliated cocycle, holonomy group, metrizability, formal integrability, transverse metric 1. Introduction Differential geometry of the total space of a manifold’s tangent bundle has its origins in diverse fields of study such as calculus of variations, differential equations, theoretical physics, and mechanics. In recent years, it can be regarded as a distinguished domain of differential geometry and has noteworthy applications in specific problems of mathematical biology and mainly in the theory of physical fields [3–5, 25–27]. This significance provides a constructive setting for the development of novel notions and geometric structures such as systems of secondorder differential equations (SODEs), metric structures, semisprays, and nonlinear connections. Accordingly, analysis of the above-mentioned concepts can be considered as a powerful tool for the thorough investigation of the geometric properties of a tangent bundle. From a historical point of view, principled investigation of the differential geometry of tangent bundles started with Dombrowski [16], Kobayashi and Nomiza [20], and Yano and Ishihara [36] in the 1960s and 1970s. Specifically, Crampin [11] and Grifone [17] considerably contributed to the geometry of the tangent bundle by introducing the notion of the nonlinear connection on the tangent bundle of a system of SODEs. In [25] Miron introduced and investigated the concept of generalized Lagrange spaces. Moreover, regarding covariant derivatives and geometric objects that can be associated to a system of SODEs, comprehensive research was undertaken in [2, 13, 21, 23, 34] (refer to [9] for more details). In the last decades increasing numbers of studies have been dedicated to the qualitative investigation of the solutions of systems of (non-)autonomous second (higher)-order ordinary (partial) differential equation fields via some corresponding geometric structures. The notable fact regarding these investigations is the significant demand for a unifying geometric setting for a differential equation field considering the associated ∗Correspondence: 81 This work is licensed under a Creative Commons Attribution 4.0 International License. AHANGARI/Turk J Math geometric structures and invariants. The inverse problem of the calculus of variations is fundamentally based on the following notable question: what are the conditions under which the solutions of a typical SODE on an m -dimensional manifold as the configuration space with the local coordinates xi d2 xi + 2Gi (x, ẋ) = 0, dt2 i ∈ {1, 2, · · · , m}, (1.1) can be regarded as the solutions of the associated Euler-Lagrange equations for some Lagrangian function L ( ) d ∂L ∂L − = 0, dt ∂xi ∂xi i ∈ {1, 2, · · · , m}. (1.2) In addition, system (1.1) can be totally characterized via a second-order vector field on the tangent bundle T M denoted by semispray. Moreover, T M is considered as the velocity space with local coordinates xi , y i := ẋi . S = yi ∂ ∂ − 2Gi (x, y) i . ∂xi ∂y (1.3) Helmholtz conditions can be regarded as one of the significant points of view to the inverse problem of the calculus of variations and are fundamentally based on the necessary and sufficient conditions for the existence of a multiplier matrix gij (x, ẋ) such that for some Lagrangian L the following identity holds: ( 2 j ) ( ) d x d ∂L ∂L i gij (x, ẋ) + 2G (x, ẋ) = − . dt2 dt ∂ ẋi ∂xi (1.4) Note that in this case the semispray S is denoted by variational or a Lagrangian vector field. Furthermore, the system (1.4) can be thoroughly reformulated as follows: ( LS ∂L i dx ∂ ẋi ) = dL, (1.5) where LS is the Lie derivative with respect to semispray S . Meanwhile, for the multiplier matrix gij , the Helmholtz conditions are illustrated by: gij = gji , ∇gij = 0, ∂gij ∂gik = , k ∂y ∂y j (1.6) gik Rjk = gjk Rik . (1.7) It is worth mentioning that conditions (1.6) can be regarded as the necessary and sufficient conditions for the existence of a Lagrange function that is defined locally and has as its Hessian the multiplier matrix gij . Likewise, conditions (1.7) demonstrate the compatibility between the given SODE structure and the multiplier matrix via some related induced geometric structures such as the Jacobi endomorphism Rji and the dynamical covariant derivative ∇ . The problem of metrizability has been investigated from several aspects in recent years. Indeed, a semispray is called metrizable if the paths of the semispray are just the geodesics of some metric space. The problem of compatibility between a system of SODEs and a metric structure on a tangent bundle has been 82 AHANGARI/Turk J Math studied by many authors and it is known as one of the Helmholtz conditions from the inverse problem of Lagrangian mechanics [1, 8, 12, 14, 15, 18, 22, 34]. The noticeable fact is that Helmholtz conditions can be totally reformulated in terms of some regular and linear partial differential operators by applying Frö licher–Nijenhuis theory as a powerful tool [10]. As a consequence, the formal integrability of the declared differential operators can be exhaustively addressed via Spencer theory (refer to [10] for more complete details). In this paper, taking into account [10], sufficient conditions for the metric associated with the semispray S are determined to extend to a transverse metric for the lifted foliated cocycle on T M . In mathematics, foliation theory can be regarded as a powerful geometric device that is applied in order to study manifolds, consisting of an integrable subbundle of the tangent bundle. In other words, a foliation locally looks like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension. Such foliations of manifolds occur naturally in various geometric fields, such as solutions of differential equations and integrable systems or in differential topology. In 1959 Reinhart introduced a particular type of foliations constructe (...truncated)