Narrow Results

We define integrable, big-isotropic structures on a manifold M as subbundles E ⊆ TM ⊕ T^*M that are isotropic with respect to the natural, neutral metric (pairing) g of TM ⊕ T^*M and are closed by Courant brackets (this also implies that [E, E^⊥_g] ⊆ E^⊥_g). We give the interpretation of such a structure by objects of M, we discuss the local geometry of the structure and we give a reduction theorem.

We prove that a pair (F = vector sub-bundle of TM, 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 sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry 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.

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as in the supergeometric setting. The review is supplemented by some new concepts and examples.