Grassmann manifold

Grassmannian

The set , , of all -dimensional subspaces in an -dimensional vector space over a skew-field . If is a field, then can be imbedded in a -dimensional projective space over as a compact algebraic variety with the aid of Grassmann coordinates (cf. Exterior algebra). In the study of the geometrical properties of a Grassmann manifold an important role is played by the so-called Schubert varieties , , which are defined as follows. If is a flag of subspaces, i.e. a chain of subspaces such that also , then

Any -dimensional algebraic subvariety in a Grassmann manifold is equivalent to a unique integer combination of the varieties , where (see [1]).

If is the field of real numbers, the field of complex numbers or the skew-field of quaternions, a Grassmann manifold over can be regarded as a compact analytic manifold (which is real if or and complex if ). These manifolds are distinguished by the fact that they are the classifying spaces for the classical groups (cf. Classical group) , and , respectively. More exactly, for any CW-complex of dimension , where 1, 2 and 4, respectively, the set of isomorphism classes of -dimensional vector bundles over with base is in a natural one-to-one correspondence with the set of homotopy classes of continuous mappings [2]. A similar theory concerning the groups and leads to the study of the Grassmann manifold ( or ) of oriented -dimensional spaces in . The manifolds listed above are closely connected, in particular, with the theory of characteristic classes (cf. Characteristic class).

The role played by Grassmann manifolds in topology necessitated a detailed study of their topological invariants. The oldest method of this study was based on Schubert varieties, with the aid of which a cell decomposition for () is readily constructed. It is found, in particular, that the cycles form a basis of the homology groups , , . Cohomology algebras of Grassmann manifolds and the effect of Steenrod powers on them have also been thoroughly studied .

Another aspect of the theory of Grassmann manifolds is that they are homogeneous spaces of linear groups over the corresponding skew-field, and represent basic examples of irreducible symmetric spaces (cf. Symmetric space).

Manifolds which are analogous to Grassmann manifolds can also be constructed from subspaces of infinite-dimensional vector spaces. In particular, an important role in the theory of deformation of analytic structures is played by a Banach analytic manifold , the elements of which are the closed subspaces of a Banach space over with a closed direct complement.

References

[1]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1952) MR0048065 Zbl 0048.14502
[2]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[3a]	A. Borel, "Sur la cohomogie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts" Ann of Math. , 57 (1953) pp. 115–207
[3b]	A. Borel, "La cohomologie mod 2 de certains espaces homogènes" Comm. Math. Helv. , 27 (1953) pp. 165–197 MR0057541 Zbl 0052.40301
[3c]	A. Borel, J.-P. Serre, "Groupes de Lie et puissances réduites de Steenrod" Amer. J. Math. , 75 (1953) pp. 409–448 MR0058213
[4]	S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004

Comments

Choose a basis in . For each choose vectors generating . These vectors generate an matrix . Now assign to the point in the projective space , , whose homogeneous coordinates are the determinants of all submatrices of . This point does not depend on the choices made. This defines an imbedding , called the Plücker imbedding. Correspondingly, these coordinates are called Plücker coordinates; they are also called Grassmann coordinates (cf. Exterior algebra and above). As a subvariety of the Grassmann manifold is given by a number of quadratic relations, called the Plücker relations, cf. [a1], Sect. 1.5.

There are a number of different notations in use; thus, the Grassmann manifold of -planes in is variously denoted (as here), , , and , the last one generalizing to with a vector space.

In the setting of algebraic geometry one defines the projective scheme defined over whose -points form .

References

[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1–2 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004