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
![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![Gr (r, V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f735affcdf205450dbdf828d766430e9965ce232)
is the set of -dimensional subspaces of . If -dimensional, one also denotes with
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![Gr (r, V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f735affcdf205450dbdf828d766430e9965ce232)
-
.
Effect of the orthogonal / unitary and linear group
In the case, the orthogonal group is effective
![O (n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a)
on through
![Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/af173d61d9b5f30ce40bbbbb38f14cc4477b0f1b)
-
.
The effect is transitive , the stabilizers are conjugated to
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.
So you get a bijection between and the homogeneous space
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.
In the case, the unitary group acts transitive and provides a bijection of the Graßmann manifold
![U.N)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9dac86141aa23bec59b25ea2c986580753b6754)
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.
Similarly, a bijection between and
is obtained for any body![\ mathbb {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1848c435e64864e9ad4efa7e46bd6bc900c35c99)
![Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/af173d61d9b5f30ce40bbbbb38f14cc4477b0f1b)
-
.
topology
As a real Graßmann manifold (of the -dimensional subspaces in ) one designates with by the identification with
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/af173d61d9b5f30ce40bbbbb38f14cc4477b0f1b)
![O (n) / O (r) \ times O (nr)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c37248ac9646e53e695a35bb13427ba2ee656c9)
given topology .
One calls a complex Graßmann manifold accordingly
![Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/af173d61d9b5f30ce40bbbbb38f14cc4477b0f1b)
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.
Canonical inclusion induces inclusion . One defines
![{\ mathbb K} ^ {n} \ subset {\ mathbb K} ^ {{n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/559a742b5d23b785a0bacff15ca03ba943e78eb1)
![Gr (r, n) \ subset Gr (r, n + 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/24a8ea7cf67990eca377ceadebcf5e54ea9c6995)
![Gr (r, \ infty): = \ lim _ {n} Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbf8bd368c82071b9e629a0744a93e33ba89c92)
as inductive Limes the one with the Limes topology .
![Gr (r, n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/af173d61d9b5f30ce40bbbbb38f14cc4477b0f1b)
Algebraic variety
Grassmann manifolds are projective varieties using Plücker embedding .
Tautological bundle
Be the projective limit with respect to the canonical inclusions and define
![{\ mathbb K} ^ {\ infty}: = \ lim _ {n} {\ mathbb K} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2fbd1387c86db187c7acadbb1f96a5ce5150ced)
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.
Then the projection onto the first factor is a vector bundle
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,
which is called a tautological or universal r-dimensional vector bundle .
Classifying figure
For every r-dimensional vector bundle there is a continuous mapping
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,
so that the pullback of the tautological bundle is under .
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![\ gamma ^ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1bb1281b5ea5c3efeb5949fef64934dc861f02)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![T_xM](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9a02a3b6f9a6808be3b99d0b27d1b97b4bb025)
![x \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df57d73e9532bb93a1439890bcddbc2806f5859)
![T_ {x} M = x + W_ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a24b91bac5133dac880b5f70435334e88fad5b7)
for a subspace . The assignment
![W_ {x} \ subset {\ mathbb R} ^ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afe7c391af7a5125293b6c5a17f25004014aff1)
![x \ rightarrow W_ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/952a189c9a92ce438a3a2a4ed2f341f966d15646)
defines a continuous mapping
![f \ colon M \ rightarrow Gr (r, m) \ subset Gr (r, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f659ee430fa5c0c2a8ada5e13a6897f20aeaef0)
and you can show that
![f ^ {*} \ gamma ^ {r} = TM](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec2b5f26075463bc0e5275b0aef1a96f7c1b611)
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:
![Gr (r, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/95364bb25574344f25ba91cc58cfb64993dab118)
![O (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5)
![GL (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd5413dd456b730c846ac98327480da21a0d8fd)
![O (r) \ rightarrow GL (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f953cd538fe93fac7dceb081935af314eca4a2)
![GL (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd5413dd456b730c846ac98327480da21a0d8fd)
![O (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5)
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.
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 .)
![V (r, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d83f4d2f88bd31d07dfdb5081a6dc9fa8a0fbb0)
![G (r, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdce1bab1efcc3efa650606e35d55993156d62a)
![O (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5)
![\ gamma ^ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1bb1281b5ea5c3efeb5949fef64934dc861f02)
![O (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5)
![O (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/307168a31698bc14ae0f0d7bccff65b128cd00d5)
![{\ mathbb R} ^ {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2703ff2e75269d9674c7fefa35f71f4e95bd56a)
The Kolimes the sequence of inclusions
![Gr (1,2) \ subset Gr (2,4) \ subset \ ldots \ subset Gr (n, 2n) \ subset \ ldots](https://wikimedia.org/api/rest_v1/media/math/render/svg/df2fb74803b2fc1cca306b971b0a45bcb7caa394)
is referred to as or . The terms are also used
![BGL ({\ mathbb K})](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d1143e067d8eae645f3455b9e49d9da214211a)
![BO ({\ mathbb K})](https://wikimedia.org/api/rest_v1/media/math/render/svg/19d7ec44ebd3d84c59f88a40f813c706f6c5d3d3)
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.
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