DIRAC STRUCTURES FROM LIE INTEGRABILITY

MIRCEA CRASMAREANU
2012 International Journal of Geometric Methods in Modern Physics (IJGMMP)  
We prove that a pair (F = vector sub-bundle of T M, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of the big tangent bundle of M . The tangent bundle in Riemannian geometry is discussed as
more » ... ry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi-Cartan-Vranceanu metrics and their Hopf bundles.
doi:10.1142/s0219887812200058 fatcat:lobdmkgi65bn3ih6lfvfgrciby