Flag structure

The same as a flag.

A flag structure of type $\nu=(n_1,\dots,n_k)$ on an $n$-dimensional manifold $M$ is a field of flags $F_x$ of type $\nu$ defined by subspaces

$$V_1(x),\dots,V_k(x)$$

of the tangent spaces $M_x$, depending smoothly on the point $x\in M$. In other words, a flag structure of type $\nu$ on $M$ is a smooth section of the bundle of flags of type $\nu$ on $M$, the typical fibre of which at the point $x\in M$ is the variety $F_\nu(M_x)$. A flag structure of type $\nu_0=(1,\dots,n-1)$ is called complete or full. A flag structure of type $\nu$ on a manifold is a $G$-structure, where $G$ is the group of all linear transformations of the $n$-dimensional vector space preserving some flag of type $\nu$. This $G$-structure is of infinite type. The automorphism group of a flag structure is, generally speaking, infinite-dimensional. The Lie algebra $L$ of infinitesimal automorphisms of a flag structure on $M$ has a chain of ideals $L_1\subset\ldots\subset L_k$, where $L_i$ consists of the vector fields $X\in L$ such that $X(x)\in V_i(x)$ for all $x\in M$.

An important special case of flag structures are those of type $(n_1)$, or $n_1$-dimensional distributions (here $k=1$, $0<n_1<n$).

A flag structure of type $\nu$ on $M$ is called locally flat, or integrable, if every point $p\in M$ has a neighbourhood $U_p$ and a coordinate system $(x^1,\dots,x^n)$ in it such that the subspace $V_i(x)$ is spanned by the vectors

$$\frac{\partial}{\partial x^1},\dots,\frac{\partial}{\partial x^{n_i}}$$

for all $x\in U_p$ and all $i=1,\dots,k$. This means that $U_p$ has a collection of foliations $S_1,\dots,S_k$ such that for all $x\in U_p$ the flag $F_x$ is defined by a collection of subspaces of $M_x$ tangent to the leaves of these foliations passing through $x$. A flag structure is locally flat if and only if for every $i=1,\dots,k$ the distribution $V_i(x)$ is involutory, that is, if for any two vector fields $X$ and $Y$ on $M$ such that $X(x)\in V_i(x)$ and $Y(x)\in V_i(x)$ for all $x\in M$, it is true that

$$[X,Y](x)\in V_i(x),$$

where $[X,Y]$ is the Lie bracket of $X$ and $Y$.

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 $(n_1,\dots,n_k)$ that is invariant relative to parallel displacements on a complete simply-connected $n$-dimensional Riemannian manifold $M$, then $M$ is isomorphic to the direct product of simply-connected Riemannian manifolds of dimensions

$$n_1,n_2-n_1,\dots,n_k-n_{k-1},n-n_k.$$

A transitive group of diffeomorphisms of a manifold $M$ leaves some flag structure of type $\nu$ on $M$ invariant if and only if its linear isotropy group preserves some flag of type $\nu$ in the tangent space to $M$. In particular, if $H$ is a closed subgroup of a Lie group $G$ such that the restriction to $H$ of the adjoint representation of $G$ gives a triangular linear group, then there is an invariant complete flag structure on the homogeneous space $G/H$, 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 $G/P$" 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