Graßmann manifolds (also Grassmann manifolds ) are a fundamental term in mathematics for both differential geometry and algebraic geometry . They parameterize the subspaces of a vector space and thus represent a generalization of the projective space . They are named after Hermann Graßmann .
definition
Let be a vector space over a body . Then designated
is the set of -dimensional subspaces of . If -dimensional, one also denotes with
-
.
Effect of the orthogonal / unitary and linear group
In the case, the orthogonal group is effective
on through
-
.
The effect is transitive , the stabilizers are conjugated to
-
.
So you get a bijection between and the homogeneous space
-
.
In the case, the unitary group acts transitive and provides a bijection of the Graßmann manifold
-
.
Similarly, a bijection between and
is obtained for any body
-
.
topology
As a real Graßmann manifold (of the -dimensional subspaces in ) one designates with by the identification with
given topology .
One calls a complex Graßmann manifold accordingly
-
.
Canonical inclusion induces inclusion . One defines
as inductive Limes the one with the Limes topology .
Algebraic variety
Grassmann manifolds are projective varieties using Plücker embedding .
Tautological bundle
Be the projective limit with respect to the canonical inclusions and define
-
.
Then the projection onto the first factor is a vector bundle
-
,
which is called a tautological or universal r-dimensional vector bundle .
Classifying figure
For every r-dimensional vector bundle there is a continuous mapping
-
,
so that the pullback of the tautological bundle is under .
In the case of the tangent bundle of a differentiable manifold , one has the following explicit description of the classifying mapping: According to Whitney's embedding theorem , one can assume that a submanifold is one . The tangent plane at a point is then of the form
for a subspace . The assignment
defines a continuous mapping
and you can show that
is.
Classifying space for principal bundles
The Graßmann manifold is the classifying space for principal bundles with structural groups . And thus also for principal bundles with a structure group , because because the inclusion is a homotopy equivalence, each bundle can be reduced to the structure group . The following applies:
-
.
The canonical projection of the Stiefel manifold to which Repere maps on each of the generated by them subspace is the universal -bundle. (The tautological bundle results from the universal bundle as an associated vector bundle through the canonical effect of on the vector space .)
The Kolimes the sequence of inclusions
is referred to as or . The terms are also used
-
.
The homotopy groups of this space can be calculated using Bott periodicity .
Schubert calculus
The cup product in the cohomology ring of the Graßmann manifolds can be determined using Schubert's calculus.
See also
Web links