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Journal of Geometric Mechanics

2015 , Volume 7 , Issue 3

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Hypersymplectic structures on Courant algebroids
Paulo Antunes and  Joana M. Nunes da Costa
2015, 7(3): 255-280 doi: 10.3934/jgm.2015.7.255 +[Abstract](37) +[PDF](335.7KB)
We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simple way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hypersymplectic structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated.
Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach
Joachim Escher and  Tony Lyons
2015, 7(3): 281-293 doi: 10.3934/jgm.2015.7.281 +[Abstract](43) +[PDF](400.4KB)
In the following we study the qualitative properties of solutions to the geodesic flow induced by a higher order two-component Camassa-Holm system. In particular, criteria to ensure the existence of temporally global solutions are presented. Moreover in the metric case, and for inertia operators of order higher than three, the flow is shown to be geodesically complete.
Lie algebroids generated by cohomology operators
Dennise García-Beltrán , José A. Vallejo and  Yurii Vorobiev
2015, 7(3): 295-315 doi: 10.3934/jgm.2015.7.295 +[Abstract](63) +[PDF](458.3KB)
By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on the tangent bundle $TM$ (and its complexification $T^{\mathbb{C}}M$) constructed from pre-existing geometric ones such as foliations, complex, product or tangent structures. We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.
Models for higher algebroids
Michał Jóźwikowski and  Mikołaj Rotkiewicz
2015, 7(3): 317-359 doi: 10.3934/jgm.2015.7.317 +[Abstract](59) +[PDF](737.7KB)
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a differential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and present some examples.
The emergence of torsion in the continuum limit of distributed edge-dislocations
Raz Kupferman and  Cy Maor
2015, 7(3): 361-387 doi: 10.3934/jgm.2015.7.361 +[Abstract](38) +[PDF](498.2KB)
We present a rigorous homogenization theorem for distributed edge-dislocations. We construct a sequence of locally-flat 2D Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion. In the process, we introduce a new notion of convergence of Weitzenböck manifolds, which is relevant to this class of homogenization problems.
A note on $2$-plectic homogeneous manifolds
Mohammad Shafiee
2015, 7(3): 389-394 doi: 10.3934/jgm.2015.7.389 +[Abstract](28) +[PDF](314.5KB)
In this note we study the existence of $2$-plectic structures on homogenous spaces. In particular we show that $S^{5}=\frac{SU(3)}{SU(2)}$, $\frac{SU(3)}{S^{1}}$, $\frac{SU(3)}{T^{2}}$ and $\frac{SO(4)}{S^{1}}$ admit a $2$-plectic structure. Furthermore, If $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $R$ is a closed Lie subgroup of $G$ corresponding to the nilradical of $\mathfrak{g}$, then $\frac{G}{R}$ is a $2$-plectic manifold.

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