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
Œ
V
}
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