Odd symplectic flag manifolds

Odd Dimensional Symplectic Manifolds

In this thesis, we introduce the odd dimensional symplectic manifolds. In the first half we study the Hodge theory on the basic symplectic manifolds. We can define two cohomology theories on them, the standard basic de Rham cohomology gheory and a basic version of the Koszul-Brylinski-Mathieu 'harmonic' symplectic cohomology theory. Among our main results are a collection of examples for which ...

Families of (1,2)-symplectic Metrics on Full Flag Manifolds

Laplacians in Odd Symplectic Geometry

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin–Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and differential forms on a purely even Lagrangian submanifold. We establish a crite...

O ct 2 00 0 Families of ( 1 , 2 ) – Symplectic Metrics on Full Flag Manifolds ∗

We obtain new families of (1,2)–symplectic invariant metrics on the full complex flag manifolds F (n). For n ≥ 5, we characterize n−3 different n–dimensional families of (1,2)–symplectic invariant metrics on F (n). Any of these families corresponds to a different class of non–integrable invariant almost complex structure on F (n).

Riemannian Symmetries in Flag Manifolds

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of Z2 -symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the Z2-symmetric structure to be naturally reductive are detailed for the flag manifold SO(5)/SO(2)× SO(2)× SO(1).