Grassmann manifolds
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Contents |
1 Introduction
Tex syntax errorbe the real, complex or quaternion field and
Tex syntax errora vector space over of dimension and let
Tex syntax error. A Grassmann manifolds of -dimensional subspaces is a set
Tex syntax errorof -dimensional subspaces. The set
Tex syntax erroris a quotient of a subset of
Tex syntax errorconsisting of linearly independent -tuples of vectors with the subspace topology. We define topology on
Tex syntax erroras the quotient topology.
Theorem 1.1 [{Milnor&Stasheff1974].}
Tex syntax erroris a Hausdorff, compact space.
Theorem 1.2 [{Milnor&Stasheff1974].}
Note that the Grassmann manifold Tex syntax erroris a connected, compact smooth manifold of dimension
Tex syntax error.
Tex syntax erroraround
Tex syntax erroris locally modelled on the vector space
Tex syntax error</div> Grassmann manifold is a homogeneous space of the general linear group. General linear group
Tex syntax erroracts transitively on
Tex syntax errorwith an isotropy group consisting of automorphisms preserving a given subspace. If the space
Tex syntax erroris equipped with a scalar product (hermitian metric resp.) then the group of isometries
Tex syntax erroracts transitively and the isotropy group of
Tex syntax erroris
Tex syntax error.
Proposition 1.3 [{Milnor&Stasheff1974].} There exist a natural diffeomorphism
The Grassmannians Tex syntax error
Tex syntax errorare projective spaces, denoted . Note that
Tex syntax error, where
Tex syntax error. If we identify
Tex syntax errorwith the one-point compactification of
Tex syntax errorthe projection corresponds to the map
Tex syntax errorgiven by
Tex syntax errorwhere
Tex syntax error. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps
Tex syntax errorfor
Tex syntax errorare called the Hopf maps and they play a very important role in homotopy theory; a fiber of
Tex syntax erroris a sphere
Tex syntax error.
Tex syntax errorin the cartesian space
Tex syntax errorwhich assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on
Tex syntax error.
Tex syntax erroron
Tex syntax errorsucht that the orbit space is homeomorphic to
Tex syntax error. Similarly for the noncompact Stiefel manifold.
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Prove that the mapTex syntax erroris locally trivial (even a principal
Tex syntax error-bundle), thus a fibration.
They are examples of coadjoint orbits [Kirillov2004]
Theorem 1.4.
2 Construction and examples
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3 Invariants
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4 Classification/Characterization
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5 Further discussion
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6 References
- [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [[Template:{Milnor&Stasheff1974|[{Milnor&Stasheff1974]]] {{{Milnor&Stasheff1974}}