Elliptic spaces in Grassmann manifolds

Full text

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
BY
JOSEeH
A.
WoL
1.
Introduction
Let
G,(F)
denote
the
Grassmann
manifold
of
n-dimensional
subspaces
of
F
,
with
its
usual
structure
as
a
Riemannian
symmetric
space,
where
F
denotes
the
real
numbers,
the
complex
numbers,
or
the
quaternions.
In
an
earlier
paper
[5]
we
studied
the
connected
totally
geodesic
submunifolds
B
of
G.(F)
with
the
property
that
any
two
distinct
elements
of
B
have
zero
intersection
as
subspaces
of
F
.
We
proved
[5,
Theorem
4]
that
B
is
iso-
metric
to
a
sphere,
to
a
real,
complex,
or
quaternionic
projective
space,
or
to
the
Cayley
projective
plane;
we
then
[5,
Theorem
8]
classified
(up
to
an
isom-
etry
of
Gn,(’))
the
manifolds
B
which
are
isometric
to
spheres.
In
Chapter
I
of
this
paper
we
show
that
B
cannot
be
the
Cayley
projective
plane
(Theorem
2),
and
we
classify
the
manifolds
B
which
are
not
isometric
to
spheres
(Theorem
3).
The
main
technique
is
the
application
of
the
results
of
the
preceding
paper
[5]
to
the
projective
lines
of
B,
which
are
totally
geodesic
spheres
in
G,(F),
resulting
in
a
structure
theorem
(Proposition
1)
for
B.
The
key
to
the
study
of
the
manifolds
B
is
the
observation
[5,
Remark
4]
that
any
two
elements
of
B
are
isoclinic
(constant
angle)
in
the
sense
of
Y.-C.
Wong
[6].
Chapter
II
is
devoted
to
the
converse
problem.
We
define
a
closure
operation
on
sets
of
pirwise
isoclinic
n-dimensionul
subspces
of
F
,
and
prove
(Lemma
10
and
Theorem
4)
that
the
closed
sets
are
finite
disjoint
unions
Bu
u
B
of
manifolds
B
where
every
element
of
B
is
orthogonal
to
every
element
of
B
(as
subspaces
of
F
)
whenever
i
j.
Thus
the
notion
"set
of
mutually
isoclinic
n-dimensional
subspaces
of
F
’’
coincides
with
the
notion
"subset
of
a
finite
union
of
mutually
orthogonal
submanifolds
B
of
G.,(F)".
As
our
structure
and
classification
theorems
completely
describe
the
manifolds
B,
this
gives
thorough
analysis
of
the
sets
of
pairwise
isoclinic
subspaces
of
any
given
dimension
in
F;
a
similar
analysis
results
for
sets
of
pairwise
Clifford-parallel
linear
subspaces
of
any
given
dimension
in
the
projective
space
P-(F).
CHAPTER
I.
THE
ELLIPTIC
SPACES
2.
Definitions
and
notation
F
will
always
denote
one
of
the
real
division
algebras
I
(real
numbers),
C
(complex
numbers),
or
K
(real
quaternions)
with
conjugation
a
--
over
I.
Given
an
integer
/
>
0,
denotes
u
hermitiun
positive-definite
left
Received
February
20,
1962.
The
author
wishes
to
thank
the
National
Science
Foundation
for
fellowship
support
during
the
period
of
preparation
of
this
pper.
447

448
JOSEPH
A.
WOLF
veetorspaee
of
dimension/c
over
F,
and
U(/,
F)
denotes
the
unitary
group
(all
linear
transformations
which
preserve
the
hermitian
structure)
of
F
U(/,
R)
is
the
orthogonal
group
O(/c)
U(k,
C)
is
the
unitary
group
U(/)
U(/c,
K)
is
the
symplectic
group
(=
unitary
symplectic
group)
Sp(k).
The
Grassmann
manifold
-n,k(),
defined
whenever
0
<
n
<
/c,
is
the
set
of
all
n-dimensional
subspaces
of
F
with
a
structure
as
Riemannian
symmet-
ric
space.
This
structure
is
defined
us
follows.
U(/c,
F)
acts
transitively
on
the
elements
of
Gn,(F)
given
B
G,,,k(F),
K
will
denote
the
isotropy
subgroup
{T
e
U(/c,
F)
T(B)
B}
of
U(k,
F)
at
B.
This
allows
us
to
identify
G,(F)
with
the
coset
space
U(lc,
F)/Kz
under
T
-+
T(B);
as
U(k,
F)
is
compact
Lie
group
and
K
is
a
closed
subgroup,
this
identifica-
tion
gives
G,(F)
the
structure
of
a
compact
analytic
munifold.
German
letters
denote
Lie
algebras,
nnd
f
is
the
Killing
form
on
lI(lc,
F).
Define
3
(relative
to
f);
then
there
is
a
vectorspce
direct-sum
de-
composition
Lt(/c,
F)
,
+
,.
This
is
a
Cartan
decomposition;
we
will
call
it
the
decomposition
of
Lt(k,
F)
at
B.
The
restriction
of
--f
to
,
is
positive-definite
and
K-invariant.
There
is
a
K
-equivariant
identification
of
3
with
the
tangentspace
to
G,(F)
at
B,
under
the
differential
of
the
projection
T
--
T(B);
thus
--f
induces
a
U(/c,
F)-invariant
Riemannian
metric
on
G,(F).
We
will
always
view
G,k(F)
with
this
Riemannian
structure.
It
is
Riemannian
symmetric,
the
symmetry
at
B
being
induced
by
the
element
of
U(/c,
F)
which
is
I
(=
identity)
on
B
and
is
-i
on
B
(=
orthogonal
complement
of
B
in
F).
If
n
1,
then
Cn,(F)
is
jUSt
a
projective
space:
where
lt(1
’)
carries
its
usual
elliptic
metric.
The
Cayley
projective
plane
l(Cay)
cannot
be
realized
this
way.
:Recall
that
a
submanifold
of
a
Riemannian
manifold
is
totally
geodesic
if
every
geodesic
of
the
submanifold
is
a
geodesic
of
the
ambient
manifold,
or,
equivalently,
if
the
submanifold
contains
every
geodesic
of
the
ambient
manifold
which
is
tangent
to
the
submanifold
at
some
point.
Let
exp
lI(/c,
F)
--
U(/c,
F)
denote
the
exponential
map.
If
(R)
is
a
subspace
of
z,
B
.
Gn,k(:F),
then
exp()(B)
is
a
totally
geodesic
submanifold
of
G.,(F)
if
and
only
if
(R)
is
a
Lie
triple
system,
i.e.,
if
and
only
if
the
Lie
product
[(R),
[(R),
(R)]]
c
(R).
For
example,
it
follows
that
the
elements
of
Gn,(F)
lying
in
a
fixed
subspace
of
Y
k
form
a
connected
totally
geodesic
submanifold.
In
particular,
the
pro-
jective
lines
of
Pt(F)
are
totally
geodesic
submanifo]ds
which
are
isometric
to
spheres;
the
same
is
true
for
1)2(Cay).
If
M
is
a
Riemannian
manifold,
then
I
(M)
denotes
the
full
group
of
isom-
etries
(self-diffeomorphisms
which
preserve
the
Riemannian
structure)
of
M.
For
example,
I(P2(Cay))
is
the
compact
exceptional
group
F4.
I0(M)
denotes
the
identity
component
of
I(M).
We
will
assume
familiarity
with
the
first
two
chapters
of
the
preceding

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
paper
[5],
and
with
the
geometry
of
the
projective
spaces
Pt(F)
and
I)2(Cay).
A
short
but
sufficient
exposition
of
p2(Cay)
can
be
found
in
[2].
3.
Geodesic
submanifolds
of
projective
spaces
We
need
to
know
the
dimensions
for
which
there
exist
totally
geodesic
spheres
in
projective
spaces
(Lemma
2).
As
it
involves
little
extra
effort,
we
will
also
derive
the
classification
of
totally
geodesic
submanifolds
in
a
Riemannian
symmetric
space
of
rank
one
(Theorem
1).
M
will
denote
a
projective
space
Pt(l)
or
P2(Cay).
IEMMA
1.
Let
N
be
a
connected
submanifold
of
M.
Then
N
is
a
totally
geodesic
submanif
old
of
M
which
is
isometric
to
a
sphere
if
and
only
if
N
is
a
totally
geodesic
submanif
old
of
a
projective
line
of
M.
Proof.
Sufficiency
is
clear
because
the
projective
lines
of
M
are
totally
geodesic
submanifolds
which
are
isometric
to
spheres.
Now
suppose
that
N
is
totally
geodesic
in
M
and
is
isometric
to
a
sphere.
Choose
x
e
N,
and
let
x
be
the
antipodal
point
of
x
on
N.
Given
y
e
N
{x,
x’},
there
is
a
unique
geodesic
,y
on
N
which
contains
x
and
y.
Observe
that
x
e
,y
and
that
,
is
contained
in
the
projective
line
Ls
of
M
determined
by
x
and
y.
Let
L
be
the
projective
line
of
M
determined
by
x
and
x;
it
follows
that
Ls
L.
ThusN
c
L.
Now
a
geodesic
of
N
is
a
geodesic
of
M
which
is
contained
in
L,
and
which
is
thus
a
geodesic
of
L.
This
shows
that
N
is
totally
geodesic
in
L,
Q.E.D.
LEMMA
2.
IV
has
a
totally
geodesic
submanifold
isometric
to
an
r-sphere
if
and
only
if
(1)
M
Pt(R)
and
r
<=
1,
(2)
M
P(C)
and
r
<=
2,
(3)
M
P(K)
and
r
<=
4,
or
(4)
M
P(Cay)
and
r
<=
8.
If
N1
and
N.
are
totally
geodesic
submanifolds
of
M
which
are
isometric
to
r-spheres,
then
I0(M)
has
an
element
which
maps
NI
onto
N
Proof.
The
first
statement
follows
from
Lemma
1
because
a
projective
line
of
M
is
a
sphere
of
dimension
1,
2,
4,
or
8,
respectively.
The
second
statement
follows
in
the
first
three
cases
from
transitivity
of
SO(t
-4-
1),
SU(t
-4-
1)
or
Sp(t
-4-
1)
on
2-dimensional
subspaces
of
R
+1,
C
t+l
or
K
+1,
respectively.
Now
let
M
P(Cay).
Applying
an
element
ef
Ic.(M)
to
N,
we
may
assume
both
N
and
N:
to
lie
in
the
same
projective
line
L
of
M,
for
I0(M)
acts
transitively
on
the
projective
lines
of
lYl.
Let
x
be
the
pole
of
L,
i.e.,
the
(unique)
focal
point
of
the
submanifold
L.
The
isotropy
subgroup
of
’
I0(M)
at
x
is
isomorphic
to
Spin(9),
the
universal
covering
group
of
the
identity
component
SO(9)
of
0(9)
U(9,
R);
it
preserves
L,
and
its

450
JOSEPH
A.
WOLF
action
on
L
is
that
of
the
usual
(linear)
action
of
SO(9)
on
S
s,
so
one
of
its
elements
carries
N1
onto
N2,
Q.E.D.
LEMMA
3.
Let
N
be
a
connected
totally
geodesic
submanif
old
of
M
which
is
not
isometric
to
a
sphere.
Then
(1)
M
pt(R)
and
N
P(R)
(2
<=
r
<-_
t),
or
(2)
M
Pt(C)
and
N
Pr(R
or
C)
(2
<=
r
<-
t),
or
(3)
M
pt(K)
and
N
pt(R,
C
or
K)
(2
<-_
r
<=
t),
or
(4)
M
P:(Cay)
and
N
P2(R,
C,
K
or
Cay).
Proof.
We
first
observe
that
pt+l(,)
cannot
be
a
totally
geodesic
sub-
pt+
L’
manifold
of
pt(l")
For
suppose
it
is.
Choose
x
e
(F),
let
L
and
be
the
respective
polars
(focal
sets)
of
x
in
pt+(,)
and
pt(,,),
and
observe
that
L
c
L’
because
Pt+(l’)
is
totally
geodesic
in
pt(l").
L
is
totally
geo-
desic
in
pt+(F),
thus
also
in
pt(F,),
thus
also
in
L’.
Now
L
pt
(F)
and
L’
pt-(,,),
so
we
have
reduced
t.
Iterating
this
procedure,
we
obtain
P(F)
as
u
totally
geodesic
submanifold
of
u
sphere
P(F’),
which
is
impossi-
ble
because
P(F)
is
not
isometric
to
a
sphere.
This
proves
r
=<
in
(1),
(2),
and
(3)
the
same
argument
proves
(4)
if
N
is
a
projective
space.
N
is
a
projective
space
because
it
is
a
Riemannian
symmetric
space
of
rank
one
which
is
not
isometric
to
a
sphere;
thus
Lemma
2
gives
the
dimensions
of
the
totally
geodesic
spheres
in
N.
Such
a
sphere
is
a
totally
geodesic
sphere
in
M.
Our
lemma
now
follows
from
Lemm
2,
Q.E.D.
:LEMMA
4.
The
inclusions
of
Lemma
3
all
exist.
Proof.
The
inclusions
of
(1),
(2),
and
(3)
obviously
exist;
thus
we
need
only
find
a
totally
geodesic
submanifold
of
P(Cay)
which
is
isometric
to
P2(K).
We
choose
[3,
p.
219]
a
maximal
subgroup
G
of
F4
I(P2(Cay))
which
is
locally
isomorphic
to
Sp(3)
X
Sp(1),
and
let
H
be
the
subgroup
of
G
for
the
local
factor
Sp(3).
G
is
normalized
by
a
symmetry
of
P(Cay),
and
this
symmetry
normalizes
H;
this
gives
x
e
P(Cay)
such
that
G(x)
and
H(x)
are
totally
geodesic
submanifolds.
G(x)
is
not
a
sphere.
For
if
it
were
a
sphere
of
dimension
>
0,
it
would
be
contained
in
a
proiective
line
L
by
Lemma
1,
and
G
would
preserve
L.
Then
G
would
leave
fixed
the
pole
of
L,
and
would
be
contained
in
an
iso-
tropy
subgroup
Spin(9)
of
F4,
contradicting
maximality
of
G
in
F.
H(x)
is
not
a
sphere.
For
H(x)
x
implies
that
H
preserves
every
element
of
G(x),
and
thus
preserves
every
projective
line
with
two
points
in
G(x).
As
G(x)
is
not
a
sphere,
H
would
preserve
many
proiective
lines,
and
would
thus
act
trivially
on
P(Cay);
this
is
impossible.
If
H(x)
is
a
sphere
of
positive
dimension,
then
H
preserves
the
projective
line
L
containing
H(x),
whence
H
preserves
the
pole
y
of
L.
G(y)
is
totally
geodesic,
so
the
preced-
ing
argument
shows
H(y)
y.
H(x)
P2(R,
C,
or
Cay).
For
equality
would
give
P(R,
C,
or
Cay)

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
451
as
a
coset
space
H/K
of
H.
Nonvanishing
of
the
Euler
characteristic
x(P2(R,
C,
or
Cay)
implies
[4,
p.
15]
that
rank
K
rank
H
3.
The
homotopy
sequence
{1}
2(H)
-+
.(P(--))
--
rl(g)
-+
I(H)
{1}
shows
(see
[2]
for
l(Cay))
that
K
has
center
of
dimension
1
for
R
or
C,
and
K
is
semisimple
for
Cay.
Now
dim
H
21,
whence
dim
K
is
19
for
R,
17
for
C,
and
5
for
Cay.
But
there
is
no
semisimple
Lie
group
of
rank
3
and
dimension
5,
nor
of
rank
2
and
dimension
16,
nor
of
rank
2
and
dimen-
sion
18.
As
H(x)
is
not
isometric
to
a
sphere,
Lemma
2
shows
that
it
is
isometric
to
1
(R,
C,
K,
or
Cay).
We
have
just
eliminated
all
except
P(K),
Q.E.D.
Let
S
denote
the
m-sphere
in
a
Riemannian
metric
of
constant
positive
curvature.
We
have
arrived
at
the
goal
of
3"
THEOREM
1.
Let
M
be
a
connected
compact
Riemannian
symmetric
space
of
ranlc
one,
and
let
N
be
a
connected
totally
geodesic
submanif
old
of
M.
Then
(1)
M
S
and
N
S
(1
<=
r
<-_
t);
or
(2)
M
I)(R),
and
either
N
S
pI(R),
or
N
I
r(R)
(2__<
r_<
t);
or
(3)
M
P(C),
and
either
N
S
(1
-_<
r
-<
2),
or
N
P(I
or
C)
(2
=<
r_-<
t);
or
(4)
M
P(K),
and
either
N
S
(1
-<
r
=<
4),
or
N
P(R,
C,
or
K)
(2
_-<
r
=<
t);
or
(5)
M
P2(Cay),
and
either
N
S
(1
<=
r
<=
8),
or
N
P(R,
C,
K,
or
Cay).
These
inclusions
all
exist;
they
are
unique
in
the
sense
that,
if
two
connected
totally
geodesic
submanifolds
of
M
are
homeomorphic,
then
they
are
equivalent
under
an
element
of
I0(M).
Proof.
By
Lemmas
2,
3,
and
4,
we
need
only
prove
the
uniqueness
when
N
is
not
a
sphere.
Now
let
N
and
N
be
connected
totally
geodesic
sub-
manifolds
of
M
Pt(R,
C,
K,
or
Cay),
Ni
P(F).
We
may
apply
an
element
of
I0(M)
to
N1,
and
assume
that
we
have
an
element
x
e
N
n
N.
Let
L,
L1,
and
L.
be
the
respective
polars
(=
focal
sets)
of
x
in
M,
and
N.
Li
is
totally
geodesic
in
Ni,
thus
in
M,
and
thus
in
L,
and
1)-I(F)
or
the
Li
are
spheres
of
the
same
dimension.
By
Lemma
2
or
in-
duction
on
t,
an
element
of
I0(L)
maps
L
onto
L..
This
element
extends
to
an
element
of
I0(M)
which
maps
N
onto
N2,
Q.E.D.
4.
Decomposition
by
projective
lines
Let
B
be
a
connected
totally
geodesic
submanifold
of
the
Grassmann
manifold
Gn,k(’)
of
n-dimensional
subspaces
of
1
k,
and
assume
that
any

452
JOSEPH
Ao
WOLF
two
distinct
elements
of
B
have
zero
intersection
as
subspaces
of
F
k
In
the
earlier
paper
[5]
we
saw
that
B
is
compact
Riemannian
symmetric
space
of
rank
one,
nd
we
classified
the
possibilities
where
B
is
a
sphere.
:Now
suppose
that
B
is
not
a
sphere;
thus
B
is
proiective
space
Pt(R,
{2,
or
K)
or
P2(Cay).
:LEMMA
5.
Choose
B
e
B,
and
let
(R)
be
the
tangentspace
to
B
at
B.
Then
there
is
an
orthogonal
direct-sum
decomposition
where
is
the
tangentspace
at
B
to
a
projective
line
L
of
B
through
B.
Remark.
Counting
dimensions,
it
is
clear
that
2
if
B
P2(Cay),
andt=
rifB
lr(R,C,
orK).
Proof.
If
B
Pr(F),
view
it
as
the
set
of
one-dimensional
subspaces
of
F+I;
we
choose
an
orthonormal
basis
{x0,
x
of
F
+1
such
that
x0
spans
B
over
F,
nd
we
define
Li
to
be
the
set
of
F-lines
in
F
+
which
lie
in
the
space
with
F-bsis
{x0,
x
}.
If
B
P(Cay),
we
choose
pro-
iective
line
L1
through
B,
we
define
B’
to
be
the
ntipodal
of
B
on
the
8-sphere
L1,
nd
we
define
L.
to
be
the
polar
of
B’
in
B;
B
L1
n
L.
because
B
is
focal
to
B’
and
L
L2.
In
either
case,
the
decomposition
of
is
easily
seen
to
be
orthogonal,
Q.E.D.
We
will
now
see
the
relation
between
the
trnsvections
of
Gn,(F)
and
the
decomposition
of
Lemma
5.
We
have
the
orthogonal
direct-sum
decomposition
lI(/c,
F)
z
of
lI(k,
F)
at
B,
under
the
Killing
form
of
lI(lc,
F),
where
K
is
the
isotropy
subgroup
of
U(k,
F)
at
B.
The
tangentspace
(R)
to
B
at
B
is
identified
as
subspace
of
.
Let
{a}
be
a
standard
basis
of
F
over
R:
a
1
--a
nd
a-
-a.ai
e
{a}
for
1
<
i
<
j.
If
x
{x,
...,
xk}
is
an
orthonormal
basis
of
F
whose
first
n
elements
span,
B,
then
recall
[5,
Chapter
II]
that
3z
has
basis
consisting
of
the
linear
:E.,)
(1
<
i
<
n
<j
<
k)
transformations
of
F
with
matrix
O/q
(Ei,.
q
relative
to
x;
given
X
e
(R),
x
can
be
chosen
such
that
the
matrix
of
X
is
a
real
multiple
of
n_--I
(E,+n
E+n,).
L
of
Lemma
5
is
an
isoclinic
sphere
on
a
2n-dimensional
subspace
V
of
F
[5,
Theorem
3],
and
it
is
clear
that
V
B
@
B
is
an
orthogonal
direct-
sum
decomposition
where
B
is
the
antipodal
of
B
on
Li.
Let
V
be
the
sub-
space
A
of
F
.
Then
we
have
LEMM
6.
V
B
@
B
@@
B
is
an
orthogonal
direct-sum
decom-
position.
Proof.
Let
x
be
an
orthonormal
basis
of
F
whose
first
n
elements
span

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
453
B,
whose
next
n
elements
span
B1,
and
in
which
an
element
X
e
1
has
matrix
1
(Ei,i+
Ei+,i).
Let
be
the
subspace
of
z
spanned
by
the
transformations
of
matrix
cq(Ei,n+j-
OqEn+j,i)
(j
n)
relative
to
x
(so
(R)1
c
),
let
B
be
the
subspace
for
j
>
n,
and
let
T"
z
-
z
be
the
transformation
Y
--
[X,
[X,
Y]].
Then
T
is
symmetric
because
X
is
skew,
T
preserves
(R),
(R)1,
and
because
they
are
Lie
triple
systems,
and
a
short
calculation
shows
that
T
induces
multiplication
by
-1
on
An
application
of
[5,
Theorem
1]
to
L1
shows
(by
the
argument
[5,
6]
that
an
isoclinic
sphere
is
totally
geodesic)
that
(R)1
is
an
orthogonal
direct
sum
IX}
-t-
(R)’1
and
T
induces
multiplication
by
-4
on
(R)’1.
As
T
is
symmetric,
there
is
an
orthogonal
direct-sum
decomposition
where
is
the
eigenspace
of
some
real
for
T.
As
T
preserves
(R)
and
it
preserves
(R)’
(R).@
@
t.
Thus(R)’
(R)’1@
(R)
:where
n.
LetY,soT(Y)
iY.
Y=
YI+
Y.withYB
and
Y
T(Y.)
--Y,
and
T(Y1)
-4Y1
by
[5,
Theorem
1
and
6]
because
it
is
readily
verified
that
every
exp(aY1)(B)
is
isoclinic
to
every
element
of
L1.
Thus
Y
e
or
Y
e
.
It
follows
from
Lemma
1
and
.
c
In
other
words,
[5,
Theorem
3]
that
Y
e
This
proves
that
(R)’
.
we
have
proved
that
B1
_1_
Bi
for
i
>
1.
Now
observe
that
the
elements
of
1
are
zero
on
Vi
Similarly,
B
_1_
B
for
i
j,
and
the
elements
of
(R)i
are
zero
on
V.
The
lemma
follows,
Q.E.D.
Lemma
6
results
in
a
good
description
of
B:
PROPOSITION
1.
Let
s
be
the
real
dimension
of
the
projective
lines
of
B.
Then
there
is
an
orthonormal
basis
x
{x,
...,
x}
of
F
such
that
{xl,
"",
x,}
is
an
orthonormal
basis
of
B
and
{x+l,
...,
x+
is
an
orthonormal
basis
of
B
(1
<=
i
-<
t),
there
is
a
basis
{X,I,
...,
X,}
of
(1
<=
i
<=
t),
and
there
are
n
X
n
F-unitary
matrices
A
(1
<=
j
<
s)
with
A
A
+
A
A
-2
I,
such
that
X.
(j
<
s)
has
matrix
0
0
A
0
0
0
0
A
0
0
0
0
0
0
0
n(i
--
1)
n(i
--
1)
k--
n(i--
1)
l
n(i
-
1)
and
X,
has
matrix

454
JOSEPH
A
WOLF
relative
to
x.
n(i+
1)
k-n(i+
1
5.
Elimination
of
the
Cayley
plane
and
the
structure
theorem
for
projective
spaces
Retain
the
notation
of
Proposition
1,
and
suppose
s
>_-
4.
Let
Yi"
(1
_-<
i
_-<
2,
1
-<_
j
=<_
s)
be
the
restriction
of
X,
to
W
B
@
B1
@
B2,
and
let
w
/wl,
w3
be
the
part
of
x
which
spans
W.
A
short
calcu-
lation
shows
that
Z
[[Y1,1,
Y2,
],
Y2,3
has
matrix
A
A
A3
0
0
0
in
ghe
basis
w
of
If.
On
ghe
other
hand,
Z
is
a
real)
-linear
combination
of
the
YI,
A
glance
at
Proposition
1
shows
that
AA
A3
4-
I.
If
s
>
4,
Proof.
Let
x0
be
an
orthonormal
basis
of
B.
We
choose
X,,
e
(R)i
and
an
orthonormalbasisxiofBisuchthattherestrictionX,lvhasmatrix(_O_i
I0)
relative
to
the
orthonormM
basis
{x0,
x}
of
V.
By
[5,
Theorems
1
and
3
and
Remark
1],
there
are
F-unitary
n
X
n
matrices
A.
(1
=<
j
<
t)
such
that
AAi
+
AAi
-2’I,
and
there
is
a
basis
{X1,1,
"",
X,}
of
(0
A0.)in
the
basis
{x0,
x}
of
V1
Let
such
that
X,.lv
has
matrix
A
Y
IX1,,,
X,,]
(1
<
i
=<
t);
the
restriction
of
Yi
to
V
+
V
B
@
B
@
B
has
mtrix
0
I
i-;
with
respect
to
the
orthonormal
basis
{xo,
xl,
x}
of
B
(9
B1
@
B.
The
ransformation
Z
[Y,
Z]
preserves
(R),
for
(R)
is
a
Lie
triple
sysgem
be-
cause
B
is
otally
geodesic;
i
sends
onto
,
onto
(R)1,
and
annihilates
he
other
summands
of
(R).
It
sends
XI,,
onto
Xi.,,
and
thus
sends
{XI.1,
XI,
onto
a
basis
{X,I
X.,}
of
(R)
X,iI
has
matrix
(.
A0)inthebasis{x0,x}ofl/’i.
Weeompletethebasis{x0,x,.-.,x}
of
B
(9
B
(9
(R)
B
to
an
orthonormal
basis
x
of
F
,
and
the
proposition
follows,
Q.E.D.

:ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
455
then
the
same
argument
shows
that
A1
A2
A
=t=I,
whence
A3
+/-A;
this
is
impossible
because
A3
and
A
anticommute.
We
have
proved
LEMMA
7.
[
Proposition
1,
either
s
1,
s
2,
or
s
4;
if
s
4,
then
A1A2
A
I.
As
an
immediate
consequence,
B
cannot
be
the
Cayley
proiective
plane,
for
s
8.
But
the
other
possibilities
for
B
exist,
subiect
to
Proposition
1
and
Lemma
7"
THEOREM
2.
Let
F
be
a
real
division
algebra,
and
let
s,
t,
n,
and
t
be
positive
integers
such
that
>=
2,
t
>__
n(t
1),ands
1,
2,
or
4.
Let
A
A8_1
be
n
X
n
F-unitary
matrices
such
that
A
A
A
A
-2
I,
and
suppose
that
A
A
A
I
in
case
s
4.
Let
x
be
an
orthonormal
basis
of
F
,
let
X.
(1
<-
i
<-
t,
1
<=
j
<=
s)
be
the
linear
transformation
of
F
with
matrix
relative
to
x
as
given
in
Proposition
1,
let
(R)
be
the
real
subspace
of
lI(]c,
F)
spanned
by
the
X,
and
define
[,
].
Then
is
a
Lie
triple
system,
so
@
is
a
subalgebra
of
ll(]c,
F).
Let
G
be
the
analytic
subgroup
of
U(]c,
F)
with
Lie
algebra
@,
and
let
B
be
the
subspace
of
F
spanned
by
the
first
n
elements
of
x.
Then
G(
B)
is
a
connected
totally
geodesic
submanif
old
of
the
Grassmann
mani-
fold
G.(F),
and
any
two
distinct
elements
of
G(B)
have
zero
intersection
as
subspaces
of
F;
G(B)
is
isometric
to
a
real
(if
s
1),
complex
(if
s
2)
or
quaternionic
(if
s
4)
projective
space
of
dimension
(topological
dimension
st).
Conversely,
if
B
is
a
connected
totally
geodesic
submanif
old
of
a
Grassmann
manifold
G,(F),
if
any
two
distinct
elements
of
B
have
zero
intersection
as
subspaces
of
F
,
and
if
B
is
not
isometric
to
a
sphere,
then
]c
3n,
and
B
is
one
of
the
manifolds
G(B)
described
above.
Proof.
Let
be
the
subspace
of
with
basis
{X.,
...,
X.8
};
(R)
.
[,
[(R)i,
]]
was
observed
in
the
proof
of
[5,
Theorem
2],
and
it
is
obvious
that
[(R),
[(R),
(R)
]]
0
if
i,
p,
and
q
are
all
different.
A
straightforward
calculation
shows
[(R),
[(R)i,
(R)
]]
(R).
By
the
Jacobi
identity,
it
follows
that
[(R),
[(R),
(R)]]
(R),
i.e.,
(R)
is
a
Lie
triple
system.
Looking
at
matrices,
we
see
that
(R)
z
where
1I(/,
F)
-
is
the
decomposition
at
B;
it
follows
that
G(B)
is
totally
geodesic
in
Let
B’
B
B
e
G(B)
#
B;
we
must
show
that
B
n
0
as
subspaces
of
F
.
G(B)
exp((R))(B);
thus
B’
exp(X)(B)
for
some
X
.
X
X
+
+
Xt,
Xi
.
,
and
we
can
conjugate
by
an
element
of
K.,
changing
basis
separately
in
each
(B),
and
assume
X
a
X.,
for
real
numbers
a.
Thus
we
may
assume
that
X
has
mtrix
0
I
a,I
0
-a
I
0
0
0
--at
I
0 0
0 0 0

456
JOSEPH
A.
WOLF
in
the
basis
x.
Now
it
is
clear,
given
b
e
B,
that
exp(X)(b)
e
B
if
and
only
if
exp(X)(b)
4-b,
and
in
that
case
exp(X)(bl)
=t=bl
for
every
b
e
B,
because
we
can
change
the
basis
of
B
without
changing
the
matrix
of
X.
Thus
either
B
B’
or,
B
n
B’
0.
It
follows
[5,
Theorem
4]
that
G(B)
is
a
real,
complex,
or
quaternionic
proiective
space,
or
the
Cayley
projective
plane.
The
remainder
of
the
theorem
follows
from
Lemma
7
and
Proposition
1,
Q.E.D.
As
any
two
distinct
elements
of
the
totally
geodesic
submanifold
G(B)
have
zero
intersection
as
subspaces
of
F
,
it
follows
[5,
Remark
4]
that
any
two
elements
of
G(B)
are
isoclinic
subspaces
of
F
.
This
leads
us
to
DEFINiTiOn.
A
submanifold
of
the
form
G(B)
in
Theorem
2
will
be
called
an
isoclinic
projective
space
on
the
subspace
of
F
with
basis
{xl
X(t+l)n}.
The
main
results
of
the
earlier
paper
with
Theorem
2
yield
[5,
Theorems
2
and
4]
combined
THEOREM
2
t.
Let
B
be
a
subset
of
Gn,(F).
Then
these
are
equivalent:
1.
B
is
a
connected
totally
geodesic
subrnanifold
of
Gn,(F),
and
any
two
distinct
elements
of
B
have
zero
intersection
as
subspaces
of
F
.
2.
B
is
an
isoclinic
sphere
on
a
2n-dimensional
subspace
of
F;
or
B
is
a
t-dimensional
(t
>=
2)
real,
complex,
or
quaternionic,
isoclinic
projective
space
on
a
-
1)
n-dimensional
subspace
of
F
.
6.
The
classification
of
isoclinic
projective
spaces
Consider
the
problem
of
existence
and
equivalence
of
the
sets
9.1’
{A1,
A8_1
of
Theorem
2.
.I’
isa
subset
of
the
F-algebra
n(F)
of
all
n
X
n
matrices
over
F;
let
n(F)R
denote
9(F)
viewed
as
an
algebra
over
R,
and
let
denote
the
subalgebra
of
gJn(F)R
generated
by
I
and
9A’.
It
is
clear
that
I
is
isomorphic
to
R
(if
s
1),
to
C
(if
s
2),
or
to
K
(if
s
4;
this
depends
on
the
fact
that
A1
A2
A3
I).
Now
let
.I1
and
I.
be
two
such
algebras,
for
the
same
F,
n,
and
s.
Except
for
the
case
s
2
and
F
C,
it
is
well
known
[1,
Theorems
4.5
and
4.14]
that
)
(F)
has
a
nonsingular
element
T
such
that
TI.
T
-
9Xl.
By
using
the
fact
that
’
and
’
generate
isomorphic
finite
subgroups
of
U(n,
F),
it
is
not
difficult
to
see
that
T
may
be
chosen
in
U(n,
F)
and
with
the
property
that
T92’
T
-
.1’.
If
we
view
T
as
a
change
of
orthonormal
basis
in
the
span
of
each
{x+,
x+.},
0
=<
i
=<
t,
then
we
have
proved
LEMMA
8.
Except
for
the
case
s
2
and
F
C,
the
manifold
G(B)
of
Theorem
2
is
determined,
up
to
a
transformation
of
U(lc,
F),
by
s,
t,
n,
to,
and
F.
In
any
case,
G(B)
exists
(i.e.,
the
A
can
be
constructed)
if
and
only
if
n
satisfies
the
condition"

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
457
F=I
F=C
F--K
s=l
s=2
no
condition
n
0
(mod
2)
no
condition
no
condition
no
condition
no
condition
s-
4:
n
0
(mod
4)
n
0
(mod
2)
no
condition
Remar]c.
The
condition
can
be
expressed"
dimR
F"
0
(mod
s).
Now
let
s
2
and
F
C.
92
c
),(C)
is
completely
determined
by
A1.
As
A1
is
unitary
with
square
-I,
it
is
unitarily
equivalent
to
a
matrix
0
--I
u+v=n.
The
nonordered
pir
{u,
v}
is
n
inwrint
of
the
unitary
equivalence
class
of
,
nd
completely
determines
that
class.
Together
with
nd
t,
{u,
v}
determines
G(B)
up
to
transformation
of
U(k,
C).
On
the
other
hnd,
in
the
terminology
of
[5,
12],
it
is
esily
seen
that
ech
projective
line
of
G(B)
is
n
isoclinic
2-sphere
of
index
{2u,
2v}
on
2n-dimensional
subspce
of
C
.
This
index
is
invrint
under
every
isometry
of
Gn,(C)
[5,
Lemm
6],
nd
is
thus
n
invrint
of
G
(B)
in
G,
(C).
DFNO.
The
index
,(G(B))
is
the
nonordered
pair
{u,
v}
in
the
dis-
cussion
aboe.
With
Lemm
8,
the
bove
discussion
yields
TnonE
3.
Consider
the
Grassmann
manifold
Gn,(F)
where
F
is
a
real
division
algebra,
and
let
F
denote
R
if
s
1,
C
if
s
2,
or
K
if
s
4.
Then
G.(F)
contains
an
isoclinic
projectie
space
Pt(F)
(t
2)
if
and
only
if
both
(t
+
1)n
k
and
dim
F
0
(rood
s).
Exceptfor
thecase
F
C
F,
any
two
isoclinic
projective
spaces
t(Fs)
in
Gn,(F)
are
equivalent
under
an
isometry
of
G,(F).
Two
isoclinic
projective
spaces
Pt(C)
in
G,(C)
are
equivalent
under
an
isometry
of
G,(
C)
if
and
only
if
they
have
the
same
index;
in
this
case
there
are
[n/2]
+
1
equivalence
classes,
the
indices
being
{0,
n},
{1,
n
1},
{In/2],
n
[n/2]},
where
denotes
integral
part.
Theorem
3
classifies
the
isoclinic
projective
spces.
Together
with
Theorem
2
nd
[5,
Theorems
4
nd
8],
it
gives
complete
description
of
the
connected
totally
geodesic
submnifolds
of
Grssmnn
mnifolds
G,(F),
for
which
ny
two
distinct
elements
of
the
submnifold
hve
zero
intersection
s
sub-
spces
of
F
.
CHwn
II.
IsocNC
ScsPcs
o
AnBWnnv
FXED
DMENSON
We
will
see
that
every
set
of
pirwise
isoclinic
n-dimensional
subspces
of
F
cn
be
enlarged
to
totally
geodesic
submnifold
of
G,(F)
in
which
ny
two
distinct
elements
hve
zero
intersection
s
subspces
of
F
.

458
JOSEPH
A.
WOLF
7.
The
closure
operation
for
isoclinic
sets
If
U
is
a
subspace
of
F
k,
then
vv
Fk
--,
U
will
denote
the
orthogonal
pro-
iection.
Recall
that
subspaces
U
and
W
of
k
are
called
isoclinic
if
the
restrictions
lw:
W
--
U
and
wl
U
--
W
are
proportional
to
unitary
transformations.
We
will
consider
only
the
case
dim
U
dim
W,
where
the
assumption
that
one
of
the
restrictions
be
proportional
to
a
unitary
transformation
automatically
forces
the
same
condition
on
the
other
restric-
tion.
Let
B
be
a
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
k.
Define
B(0)
B,
and
suppose
that
we
have
constructed
the
sequence
of
sets
of
pairwise
isoclinic
n-dimensionM
subspaces
of
.
Given
distinct
nonorthogonM
elements
B
and
B’
of
B(,
let
Si,z.z,
be
the
isoclinic
sphere
on
B
@
B’
constructed
as
in
[5,
Chapter
I]
from
the
set
of
M1
elements
of
B
which
lie
in
B
B’.
The
elements,
of
B
re
pirwise
isoclinic,
s
re
the
elements
of
S,,,.
Now
let
X
e
B()
and
Y
S,,z,.
z[x
and
z,[x
are
proportional
to
unitary
maps;
it
follows
that
either
Z
ze.,(X)
0,
or
that
dim
Z
n
and
X
is
isoclinic
to
a
subspace
of
B
@
B’
if
and
only
if
Z
is
isoclinic
to
that
subspuce.
Suppose
dim
Z
n.
Now
Z
is
isoclinic
to
every
element
of
B()
lying
in
B
@
B’;
it
follows
from
[5,
Theorem
1]
that
Z
is
isoclinic
to
every
element
of
S,.,z,.
Thus
X
and
Y
are
isoclinic.
We
have
just
proved
that
the
elements
of
B()
u
S,.,,
are
pairwise
isoclinic.
Define
B(+)
B()
u
{,,}
Si,,,
where
{B,
B
}
runs
over
M1
pirs
of
distinct
nonorthogonl
elements
of
B).
If
{B,
B’}
nd
{A,
A’}
are
two
such
pirs,
then
substitution
of
B)
u
S..,
for
B()
in
the
bove
argument
shows
that
the
elements
of
B(+)
re
pirwise
isoclinic.
Thus
we
hve
constructed
sequence
of
sets
of
pairwise
isoclinic
n-dimensional
subspaces
of
F
.
DEFINITION.
The
isoclinic
closure
B,
of
B
is
defined
by
B,
U0
B
is
said
to
be
isoclinically
closed
if
B
B,.
This
definition
is
iustified
by
LEMMA
9.
Let
B
be
a
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
and
let
B,
be
its
isoclinic
closure.
Then
B,
is
an
isoclinically
closed
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
.
Proof.
Choose
B
and
B’
in
B,.
They
lie
in
some
B(),
and
are
thus
isoclinic.
This
proves
that
the
elements
of
B,
are
pairwise
isoclinic.
Let
B,
(B,)(0).
We
must
prove
that
(B,)()
(B,)(0).
It
will
follow
that
B,
(B,),,
proving
B,
to
be
isoclinically
closed.
Let
B
and

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
459
B’
be
distinct
nonorthogonal
elements
of
B,,
let
A
be
the
collection
of
all
elements
of
B,
which
lie
in
B
@
B’,
and
let
S
be
the
isoclinic
sphere
on
B
(R)
B’
constructed
from
A
as
in
[5,
Theorem
1].
We
must
prove
that
S
c
B,;
it
will
follow
that
(B,)
(0)
(B,)
(1)
As
A
c
B,,
it
sufSces
to
prove
S
A.
For
this,
we
need
only
prove
that
A
is
an
isoclinic
sphere
on
B
(R)
B’.
Let
A,
A’
e
A.
For
some
integer
m,
B(m)
contains
B,
A,
and
A’.
Thus
B(m+l)
contains
an
isoclinic
sphere
on
B
@
B’
which
contains
A
and
A’.
It
follows
that
A
is
an
isoclinic
sphere
on
B
@
B’,
Q.E.D.
8.
The
notion
of
reducibility
for
isoclinic
sets
Let
B
be
a
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
F
k.
Given
B,
B’
B’
e
B,
we
say
B
if
there
is
a
sequence
{B
B,,
B,
Bm
B’}
in
B
such
that
B+
is
not
orthogonal
to
B.
This
is
easily
seen
to
be
an
equiv-
alence
relation
on
B.
DEFINITION.
The
equivalence
classes
in
B
will
be
called
the
irreducible
components
of
B.
B
will
be
called
irreducible
if
it
has
just
one
equivalence
class.
Given
B
e
B,
the
equivalence
class
of
B
will
be
called
the
irreducible
component
of
B
in
B.
DEFINITION.
The
support
supp
B
of
B
is
the
subspace
of
F
spanned
by
the
union
of
the
elements
of
B.
Suppose
B
e
B
B’,
where
B’
is
a
set
of
pairwise
isoclinic
subspaces
of
F
.
If
A
and
A’
are
the
respective
irreducible
components
of
B
in
B
and
B’,
then
it
is
clear
that
A
c
A’
and
thus
supp
A
suppA’.
Our
definitions
are
justified
by
].EMMA
10.
Let
B
be
a
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
,
and
let
B,
be
its
isoclinic
closure.
Then
B
and
B,
have
finite
and
consistent
decompositions
B,=BuB,u-..uB
UU
U
U
B
B
u
B
u
u
B
into
irreducible
components,
and
B,
is
the
isoclinic
closure
of
B
i.
If
i
j,
then
supp
B
supp
B,
I_
supp
B,
supp
B
.
If
we
topologize
B,
as
a
subset
of
the
Grassmann
manifold
G,(F),
then
its
connected
components
are
precisely
its
irreducible
components.
Proof.
If
two
elements
of
B
are
not
orthogonal,
then
they
lie
in
the
same
irreducible
component
of
B;
it
follows
that
distinct
irreducible
components
of
B
have
supports
orthogonal
to
each
other.
By
finite-dimensionality
of
F
,
B
has
only
a
finite
number
of
irreducible
components.
Let
B
BuB2u
uB
be
the
decomposition
of
B
into
its
irreducible
components.

460
JosP
A.
WOLF
It
is
clear
from
7
that
B,
(B1),
u
(B2),
u
u
(Bin),,
that
each
supp
B
supp
(B),,
and
that
each
(B),
is
irreducible.
Setting
B
(B),,
the
consistent
decomposition
follows
easily,
as
does
orthogonality
of
supports.
The
orthogonality
of
supports
shows
that,
in
the
topology
on
B,
induced
by
Gn.k(’),
each
B
is
a
closed
subset
of
B,.
Thus
we
need
only
prove
that
each
B
is
a
connected
subset
of
Gn.k(l).
Let
B,
B’
e
B,.
As
B,
is
irreducible,
we
have
a
sequence
{B
B1, B2,
...,
B
B’}
B,
such
that
B+I
is
not
orthogonal
to
B
(1
-<
u
<
t).
Let
Su
{B"
e
],
B
*
C
Bu
@
Bu+l
},
S
is
an
isoclinic
sphere,
thus
homeomorphic
to
a
sphere
[5,
Theorem
2];
it
follows
that
S
contains
an
arc
from
B
to
B+I.
Joining
these
arcs,
we
have
proved
that
B
is
arcwise
connected.
Thus
B2
is
connected,
Q.E.D.
9.
Isoclinic
sets
as
submanifolds
of
Grassmcnn
manifolds
The
main
result
of
Chapter
II,
sort
of
converse
to
Theorem
2’,
is
THEOREM
4.
Let
B
be
an
irreducible
isoclinically
closed
set
of
pairwise
isoclinic
n-dimensional
subspaces
of
F
,
where
F
is
a
real
division
algebra,
and
view
B
as
a
subset
of
the
Grassmann
manifold
Gn,(F).
Then
B
is
a
connected
totally
geodesic
submanifold
of
Gn.(F)
in
which
any
two
distinct
elements
have
zero
intersection
as
subspaces
of
F
.
In
view
of
Lemma
10,
it
suffices
to
prove
that
B
is
a
totally
geodesic
sub-
manifold
of
G.
(F).
Proof.
Choose
B
e
B,
let
lI(k,
’)
&
+
z
be
the
decomposition
of
ll(k,
’)
at
B,
and
define
open
neighborhoods
v
B’
B’},
V--{B’
eB"
B’
CB’}
VnS
of
B
in
G.(I)
gnd
in
B.
We
define
(R)
{Xez’exp(tX)(B)
eC
for
-1
=<
t-<
1}
nd
observe
that
U
exp
((R))
(B).
Let
be
the
real
subspace
of
z
spanned
by
(R).
If
we
can
prove
that
(R)
contains
a
neighborhood
of
ero
in
,
then
it
will
follow
that
B
is
a
regularly
imbedded
submanifold
of
G.k(’)
and
that
is
the
tangentspace
to
B
gt
B.
When
this
is
done,
suppose
B
B’
e
B,
B’
lying
in
a
normal
coordinate
neighborhood
of
B
in
V.
B’
exp
(X)(B)
for
some
X
e
(R),
nd
{exp
(tX)(B)
"t
e
R}
is
the
minimizing
geodesic
in
Gn.(’)
between
B
and
B’.
On
the
other
hnd,
it
is
an
isoclinic
1-sphere
on
B
@
B’,
nd
is
thus
contained
in
B
because
B
is
isoclinically
closed.
It
follows
that the
submanifold
B
is
totally
geodesic.

ELLIPTIC
SPACES
IN
GRASSMANN
MANIFOLDS
Ax6[
Let
X
and
Y
be
elements
of
(R).
Given
small
e
R,
we
will
prove
that
exp
(t(X
-+-
Y))(B)
e
B.
This
suffices
to
show
that
(R)
contains
a
neighbor-
hood
of
zero
in
,
proving
the
theorem.
Define
B1
exp
(X)(B)
and
B2
exp
(Y)(B).
If
dim
(B
-t-
B1
-t-
B2)
-<
2n,
then
F
has
a
2n-dimen-
sional
subspace
V
which
contains
every
exp
(tX)
(B)
and
every
exp
(tY)
(B).
The
elements
of
B
which
lie
in
V
form
an
isoclinic
sphere
A,
for
B
was
as-
sumed
isoclinically
closed.
As
X
and
Y
are
tangent
to
A
at
B,
and
as
A
is
a
totally
geodesic
submanifold
of
Gn.(F)
[5,
Theorem
2],
it
follows
that
exp
(t(X
+
Y))(B)
e
A
c
B
for
every
real
t.
Thus
we
may
assume
that
dim
(B
+
B
+
B)
>
2n.
Let
W
B
-
B
+
B.
B
has
no
nonzero
element
in
common
with
any
exp
(tX)
(B)
it
follows
that
dim
W
3n.
We
may
choose
an
orthonormal
basis
w
{w,
Wn
Of
W,
whose
first
n
elements
span
B,
such
that
the
restriction
X]
has
matrix
a_,=
(E,+
E+n.
where
0
<
a
<
1.
Let
B’
exp
(a-X)(B);
define
B"
exp
(fl-Y)(B)
similarly.
If
B’
+/-
B’,
then
we
may
assume
that
w
was
chosen
such
that
YI
has
matrix
15E=l
(Ei,i+2n
Ei+2n,i
).
A
short
calculation
shows
that
exp
(t(X
+
Y))(B)
lies
in
the
isoelinie
1-
sphere
determined
by
exp
(X/
tX)
(B)
and
exp
(%/
Y)
(B),
for
small
t,
and
is
thus
contained
in
B.
If
B
is
not
orthogonal
to
B’,
we
examine
the
isoclinic
l-sphere
determined
by
exp
(X)(B)
and
exp
(Y)(B)
(
small).
It
has
an
element
B3
such
that
W
B
+
B
+
B3
and
B3
exp
(Z)(B)
where
Z
e
(R)
and
the
B’"
exp
(,-1Z)(B)
(defined
in
the
same
way
as
B’)
is
orthogonal
to
B’.
X
+
Y
rX
+
rZ
for
some
real
numbers
and
r;
from
the
case
B’
+/-
B",
it
follows
that
exp
(t(X
+
Y))(B)
e
B
for
small
t,
Q.E.D.
10.
Summary
We
summarize
the
results
of
this
paper
and
the
earlier
one
[5].
If
B
is
a
connected
totally
geodesic
submanifold
of
the
Grassmann
mani-
fold
Gn.(’),
and
if
any
two
distinct
elements
of
B
have
zero
intersection
as
subspaces
of
F
,
then
1.
B
is
isometric
to
a
sphere
(these
manifolds
are
described
in
[5,
Theorem
1]
and
classified
in
[5,
Theorem
8])
or
to
a
real,
complex,
or
quaternionic
projective
space
(these
manifolds
are
described
in
Theorem
2
and
classified
in
Theorem
3).
2.
B
is
an
irreducible
isoelinically
closed
set
of
pairwise
isoclinie
n-dimen-
sional
subspaces
of
F
.
If
A
is
a
set
of
pairwise
isoelinic
n-dimensional
subspaces
of
F
,
then
there
is
an
orthogonal
direct-sum
decomposition
F
V1
(R)
@
Vm,
and
there
are
connected
totally
geodesic
submanifolds
B
of
Gn.(’),
such
that
every
element
of
B
lies
in
Vi,
any
two
distinct
elements
of
B
have
zero
intersec-

462
JOSEPH
A.
WOLF
tion
as
subspaces
of
F
k,
and
A
c
[3
B
B
=1
Here
[3__1
is
the
isoclinic
closure
of
A.
This
gives
a
complete
analysis
of
the
sets
of
pairwise
isoclinic
subspaces
of
any
given
dimension
in
F
k,
which,
in
turn,
gives
a
complete
analysis
of
the
sets
of
Clifford-parallel
linear
subspaces
of
any
given
dimension
in
P-I(F).
REFERENCES
1.
A.
A.
ALBERT,
Structure
of
algebras,
Amer.
Math.
Soc.
Colloquium
Publications,
vol.
24,
1939.
2.
A.
BOREL,
Le
plan
projectif
des
octaves
et
les
spheres
comme
espaces
homognes,
C.
R.
Acad.
Sci.
Paris,
vol.
230
(1950),
pp.
1378-1380.
3.
A.
BOREL
AND
J.
DE
SIEBENTHAL,
Les
sous-groupes
ferms
de
rang
maximum
des
groupes
de
Lie
clos,
Comment
Math.
Helv.,
vol.
23
(1949),
pp.
200-221.
4.
H.
SAMELSON,
On
curvature
and
characteristic
of
homogeneous
spaces,
Michigan
Math.
J.,
vol.
5
(1958),
pp.
13-18.
5.
J.
A.
WO,F,
Geodesic
spheres
in
Grassmann
manifolds,
Illinois
J.
Math.,
vol.
7
(1963),
pp.
425-446.
6.
Y.-C.
WoNH,
Isoclinic
n-planes
in
Euclidean
2n-space,
Clifford
parallels
in
elliptic
(2n
1)-space,
and
the
Hurwitz
matrix
equations,
Mem.
Amer.
Math.
Soc.,
vol.
41,
1961.
THE
INSTITUTE
FOR
ADVANCED
STUDY
PRINCETON,
NEW
JERSEY