View source for Flag structure

The same as a [[Flag|flag]].

A flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405702.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405703.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405704.png" /> is a field of flags <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405705.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405706.png" /> defined by subspaces

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405707.png" /></td> </tr></table>

of the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405708.png" />, depending smoothly on the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f0405709.png" />. In other words, a flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057011.png" /> is a smooth section of the bundle of flags of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057013.png" />, the typical fibre of which at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057014.png" /> is the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057015.png" />. A flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057016.png" /> is called complete or full. A flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057017.png" /> on a manifold is a [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057018.png" />-structure]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057019.png" /> is the group of all linear transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057020.png" />-dimensional vector space preserving some flag of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057021.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057022.png" />-structure is of infinite type. The automorphism group of a flag structure is, generally speaking, infinite-dimensional. The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057023.png" /> of infinitesimal automorphisms of a flag structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057024.png" /> has a chain of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057026.png" /> consists of the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057029.png" />.

An important special case of flag structures are those of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057030.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057032.png" />-dimensional distributions (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057034.png" />).

A flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057036.png" /> is called locally flat, or integrable, if every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057037.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057038.png" /> and a coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057039.png" /> in it such that the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057040.png" /> is spanned by the vectors

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057042.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057043.png" />. This means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057044.png" /> has a collection of foliations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057045.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057046.png" /> the flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057047.png" /> is defined by a collection of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057048.png" /> tangent to the leaves of these foliations passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057049.png" />. A flag structure is locally flat if and only if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057050.png" /> the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057051.png" /> is involutory, that is, if for any two vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057056.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057057.png" />, it is true that

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057059.png" /> is the [[Lie bracket|Lie bracket]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057061.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057062.png" /> that is invariant relative to parallel displacements on a complete simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057063.png" />-dimensional Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057064.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057065.png" /> is isomorphic to the direct product of simply-connected Riemannian manifolds of dimensions

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A transitive group of diffeomorphisms of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057067.png" /> leaves some flag structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057068.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057069.png" /> invariant if and only if its linear isotropy group preserves some flag of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057070.png" /> in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057071.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057072.png" /> is a closed subgroup of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057073.png" /> such that the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057074.png" /> of the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057075.png" /> gives a triangular linear group, then there is an invariant complete flag structure on the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057076.png" />, and also an invariant flag structure of every other type.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.N. Bernshtein, I.M. Gel'fand, S.I. Gel'fand, "Schubert cells and cohomology of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040570/f04057077.png" />" ''Russian Math. Surveys'' , '''28''' : 3 (1973) pp. 1–26 ''Uspekhi Mat. Nauk'' , '''28''' : 3 (1973) pp. 3–26 {{MR|0686277}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> K. Kodaira, D.C. Spencer, "Multifoliate structures" ''Ann. of Math.'' , '''74''' (1961) pp. 52–100 {{MR|0148086}} {{ZBL|0123.16401}} </TD></TR></table>