Flag structure

The same as a flag.

A flag structure of type on an -dimensional manifold is a field of flags of type defined by subspaces

of the tangent spaces , depending smoothly on the point . In other words, a flag structure of type on is a smooth section of the bundle of flags of type on , the typical fibre of which at the point is the variety . A flag structure of type is called complete or full. A flag structure of type on a manifold is a -structure, where is the group of all linear transformations of the -dimensional vector space preserving some flag of type . This -structure is of infinite type. The automorphism group of a flag structure is, generally speaking, infinite-dimensional. The Lie algebra of infinitesimal automorphisms of a flag structure on has a chain of ideals , where consists of the vector fields such that for all .

An important special case of flag structures are those of type , or -dimensional distributions (here , ).

A flag structure of type on is called locally flat, or integrable, if every point has a neighbourhood and a coordinate system in it such that the subspace is spanned by the vectors

for all and all . This means that has a collection of foliations such that for all the flag is defined by a collection of subspaces of tangent to the leaves of these foliations passing through . A flag structure is locally flat if and only if for every the distribution is involutory, that is, if for any two vector fields and on such that and for all , it is true that

where is the Lie bracket of and .

The existence of global (everywhere-defined) flag structures on a manifold imposes fairly-strong restrictions on its topological structure. For example, there is a line field, that is, a flag structure of type , on a simply-connected compact manifold if and only if its Euler characteristic vanishes. There is a complete flag structure on a simply-connected manifold if and only if it is completely parallelizable, that is, if its tangent bundle is trivial. If there is a parallel flag structure of type that is invariant relative to parallel displacements on a complete simply-connected -dimensional Riemannian manifold , then is isomorphic to the direct product of simply-connected Riemannian manifolds of dimensions

A transitive group of diffeomorphisms of a manifold leaves some flag structure of type on invariant if and only if its linear isotropy group preserves some flag of type in the tangent space to . In particular, if is a closed subgroup of a Lie group such that the restriction to of the adjoint representation of gives a triangular linear group, then there is an invariant complete flag structure on the homogeneous space , and also an invariant flag structure of every other type.

A theory of deformations of flag structures on compact manifolds has been developed [4].

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[3]	I.N. Bernshtein, I.M. Gel'fand, S.I. Gel'fand, "Schubert cells and cohomology of the spaces " Russian Math. Surveys , 28 : 3 (1973) pp. 1–26 Uspekhi Mat. Nauk , 28 : 3 (1973) pp. 3–26 MR0686277
[4]	K. Kodaira, D.C. Spencer, "Multifoliate structures" Ann. of Math. , 74 (1961) pp. 52–100 MR0148086 Zbl 0123.16401