On the explicit expressions of the canonical 8-form on Riemannian manifolds with Spin (9) holonomy

6 pags. 1991 Mathematics Subject Classification. Primary 53C29, Secondary 53C27.Two explicit expressions of the canonical 8-form on a Riemannian manifold with holonomy group Spin(9) have been given: One by the present authors and another by Parton and Piccinni. The relation between these two expressions is obtained. Moreover, it is shown that they are different only from a combinatorial viewpoint.The first author has been supported by DGI (Spain) Project MTM2013-46961-P..Peer reviewe

The canonical 8-form on manifolds with holonomy group Spin(9)

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained

On the cohomology of some exceptional symmetric spaces

This is a survey on the construction of a canonical or "octonionic K\"ahler" 8-form, representing one of the generators of the cohomology of the four Cayley-Rosenfeld projective planes. The construction, in terms of the associated even Clifford structures, draws a parallel with that of the quaternion K\"ahler 4-form. We point out how these notions allow to describe the primitive Betti numbers with respect to different even Clifford structures, on most of the exceptional symmetric spaces of compact type.Comment: 12 pages. Proc. INdAM Workshop "New Perspectives in Differential Geometry" held in Rome, Nov. 2015, to appear in Springer-INdAM Serie

Lagrangian reductive structures on gauge-natural bundles

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.Comment: 11 pages, remarks and comments added, this version published in ROM

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L ∇ variationally equivalent to the second-order Einstein- Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇

Poisson–Poincaré reduction for Field Theories

Given a Hamiltonian system on a fiber bundle, there is a Poisson covariant formulation of the Hamilton equations. When a Lie group G acts freely, properly, preserving the fibers of the bundle and the Hamiltonian density is G-invariant, we study the reduction of this formulation to obtain an analogue of Poisson–Poincaré reduction for field theories. This procedure is related to the Lagrange–Poincaré reduction for field theories via a Legendre transformation. Finally, an application to a model of a charged strand evolving in an electric field is given

Routh reduction for singular Lagrangians

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian $L$ or the momentum map $J_L$ are required apart from the momentum being a regular value of $J_L$. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler-Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler-Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.Comment: 34 pages, 2 figures, accepted for publicaton in International Journal of Geometric Methods in Modern Physics (IJGMMP

On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.Comment: 4 figures. Added some comments about total curvature and other detail