Graphics Reference
In-Depth Information
It is clear that g n is a vector bundle because the sets U in (3) form an open cover
of P n .
The canonical line bundle g n over P n is nontrivial.
8.13.1. Theorem.
Proof.
The proof is the same as the proof for g 1 in Example 8.9.2.
8.14
The Grassmann Manifolds
Definition. Let n > 0 and k ≥ 0. The Stiefel manifold or Stiefel variety V n ( R n+k )
is defined to be the subspace of ( S n+k-1 ) n consisting of all n-tuples ( u 1 , u 2 ,..., u n ),
where the u i are an orthonormal set of vectors of R n+k . The Grassmann manifold
or Grassmann variety or Grassmannian G n ( R n+k ) is defined to be the set of n-dimen-
sional linear subspaces of R n+k
with the quotient topology induced by the projection
map
(
) Æ
(
)
p nn nk
+
nk
+
:
V
R
G
R
n
defined by
p n
(
(
uu
,
,...,
u
)
) =
plane spanned by the
u
.
12
n
i
Stiefel and Grassmann manifolds are, like the projective spaces P n , very impor-
tant spaces in topology. They have been studied extensively. Here are a few facts about
them:
(1) The pair (V n ( R n+k ),p n ) is a locally trivial bundle over G n ( R n+k ) with all fibers
homeomorphic to O (n).
(2) The map that sends an n-dimensional linear subspace of R n+k to its k-
dimensional orthogonal complement defines a natural homeomorphism
between G n ( R n+k ) and G k ( R n+k ).
(3) There is a canonical n-dimensional vector bundle g n ( R n+k ) = ( E ,p,G n ( R n+k ))
over G n ( R n+k ), where
E = (( V , v ) | V is an n-dimensional linear subspace of R n+k and v ΠV }
à G n ( R n+k ) ¥ R n+k
and
p( V , v ) = V .
(4) P n = G 1 ( R n+1 ) and g n =g 1 ( R n+1 ). In other words, the Grassmann manifolds can
be thought of as generalizations of projective space.
(5) The n-plane bundles g n ( R n+k ) play a fundamental role in the classification of
vector bundles over a space. Let B be a topological space and let Vect n ( B )
denote the isomorphism classes [x] of n-plane bundles x over B . Define a map
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