View source for Grassmann manifold

'' Grassmannian''

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450302.png" />, of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450303.png" />-dimensional subspaces in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450304.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450305.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450306.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450307.png" /> is a field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450308.png" /> can be imbedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g0450309.png" />-dimensional projective space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503010.png" /> as a compact algebraic variety with the aid of Grassmann coordinates (cf. [[Exterior algebra|Exterior algebra]]). In the study of the geometrical properties of a Grassmann manifold an important role is played by the so-called Schubert varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503012.png" />, which are defined as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503013.png" /> is a [[Flag|flag]] of subspaces, i.e. a chain of subspaces such that also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503014.png" />, then

<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/g/g045/g045030/g04503015.png" /></td> </tr></table>

Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503016.png" />-dimensional algebraic subvariety in a Grassmann manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503017.png" /> is equivalent to a unique integer combination of the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503019.png" /> (see [[#References|[1]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503020.png" /> is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503021.png" /> of real numbers, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503022.png" /> of complex numbers or the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503023.png" /> of quaternions, a Grassmann manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503024.png" /> can be regarded as a compact analytic manifold (which is real if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503025.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503026.png" /> and complex if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503027.png" />). These manifolds are distinguished by the fact that they are the classifying spaces for the classical groups (cf. [[Classical group|Classical group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503030.png" />, respectively. More exactly, for any [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503031.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503033.png" /> 1, 2 and 4, respectively, the set of isomorphism classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503034.png" />-dimensional vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503035.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503036.png" /> is in a natural one-to-one correspondence with the set of homotopy classes of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503037.png" /> [[#References|[2]]]. A similar theory concerning the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503039.png" /> leads to the study of the Grassmann manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503040.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503041.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503042.png" />) of oriented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503043.png" />-dimensional spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503044.png" />. The manifolds listed above are closely connected, in particular, with the theory of characteristic classes (cf. [[Characteristic class|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503046.png" />) is readily constructed. It is found, in particular, that the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503047.png" /> form a basis of the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503050.png" />. 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503051.png" />, the elements of which are the closed subspaces of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503052.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503053.png" /> with a closed direct complement.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''2''' , Cambridge Univ. Press (1952) {{MR|0048065}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> A. Borel, "La cohomologie mod 2 de certains espaces homogènes" ''Comm. Math. Helv.'' , '''27''' (1953) pp. 165–197 {{MR|0057541}} {{ZBL|0052.40301}} </TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> A. Borel, J.-P. Serre, "Groupes de Lie et puissances réduites de Steenrod" ''Amer. J. Math.'' , '''75''' (1953) pp. 409–448 {{MR|0058213}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR></table>



====Comments====
Choose a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503054.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503055.png" /> choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503056.png" /> vectors generating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503057.png" />. These vectors generate an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503058.png" /> matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503059.png" />. Now assign to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503060.png" /> the point in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503062.png" />, whose homogeneous coordinates are the determinants of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503063.png" /> submatrices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503064.png" />. This point does not depend on the choices made. This defines an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503065.png" />, called the Plücker imbedding. Correspondingly, these coordinates are called [[Plücker coordinates|Plücker coordinates]]; they are also called Grassmann coordinates (cf. [[Exterior algebra|Exterior algebra]] and above). As a subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503066.png" /> the Grassmann manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503067.png" /> is given by a number of quadratic relations, called the Plücker relations, cf. [[#References|[a1]]], Sect. 1.5.

There are a number of different notations in use; thus, the Grassmann manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503068.png" />-planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503069.png" /> is variously denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503070.png" /> (as here), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503072.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503073.png" />, the last one generalizing to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503074.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503075.png" /> a vector space.

In the setting of algebraic geometry one defines the projective scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503076.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503077.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503078.png" />-points form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045030/g04503079.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR></table>