Pub
.
Ma
.
UAB
Vol
.
26
N2
3
Des
.
1982
IMMERSING
HOMOGENEOUS
SPACES
IN EUCLIDEANSPACE
Howa d
Hille *
In [4],
Lam
p o es,
amongo he
hings,
an imme sion
e-
sul
o
he
eal lag
mani old
G
IR
(ni
1
. .
.,n
S
)

=
0(n1+
. .
.+ns)/0(n1)x
. .
.x0(nS)
whe e
0(n)
is
he
ealo hogonalg oup
.
His
esul
is
a
specialcaseo
he
ollowingmo egene al
obse a ion
.
Pnopobí íon
1
.
Le
G
be
a
compac ,
connec ed
semisimple
Lie
g oup
and
H a
closedsubg oup
.
Then
ei he
G/H
is
a
i -mani old
(anal
so
imme ses
in
codimension
one) o
G/H
imme _
ses
in
~dim,(S)
whe e

is
heLie
algeb a
o
G
.
Rema k
2
.
The
dimension
o
he
ambien
Euclidean
space
is
independen
o
he
subg oup
H,
so
one
expec s
he
s onges
esul s
o
small
H
.
Bu
i
H
is
a
maximal o us
(all
n
i
=
1
in
abo e
example) hen
G/H
is
a
7 -mani old
[1]
.
See
Rema k
4
below
.
P coob
.
A
eal ec o bundle
o e
G/H
is
de e minedby
an ac ion
o
H
on
a
eal ec o space
.
The
angen
bundle
T(G/H)

comes
om
he
adjoin
ac ion
o
H
on
g/~,g
=
Lie
EL
Le
n
deno e
he
bundle
o e
'
G/H
coming om
he
adjoin
ac ion
o
H
on
B
.
C ea ly
n
is
i ial
since
he
ac ion
ex ends
o
all
o
G
.
The e
is
a
bundle
epimo phism
n-
.T(G/H)
*
Suppo ed
by
he
Alexande
onHumbold -S i ung
.
43
which
necessa ily
spli s
(see
o
example
5,p
.461)
.
Since
dim(71)
=
dim(9),
he
heo em
o
Hi sch
131
yields
he
desi ed
imme sion
.
(2)
in R

;
whe e
n
=
nl+
..
.+ns'
Cono
.21any
3
.

(Lam)

The
mani old

GIR(ni,
.
.
.
.
ns)
Pxoo4
.
Obse e
dim
0(n)
=
(2)
.
Cono
.elany
5
.
These esui s
a e
aao
as s ong
a

heseo
Lam
(41
.
.
inme ses
Remank
4
.
Tha hese
imme sions
a e
eally
in e es ing
ollows
om
[21

whe e
i
was
shown ha
i s
=
2,
ni
=
n2,
one
ob ains
a
bes possible
imme sion
o
he
eal
G assmannian
.
Simila ly,
one
ob ains
in
he
complex
and
qua e nioniccases
.
12
(a)
Ga(nl,
. .
.,ns)
imme ses
in
IR
n
(b)
GIH(nl,
.
.
.,ns)
imme ses
in
IR
2 2
n
+
n
REFERENCES
1
.
A
.
Bo el
and
F
.
Hi zeb uch,
Cha ac e is ic
classes
and
homogeneous
spaces,
III,
Ame
.
J
.
Ma h
.
.82(1960)491-504
.
2
.
H
.
Hille
and
R
.
S ong
.
Imme sion
dimension
o
eal
G assmannians,Ma h
.
Ann
.
255(1981),
361-367
.
3
.
M
.
Hi sch,
Imme sions
o mani olds,T ans
.
Ame
.
Ma h
.
Soc
.
93(1959),
242-276
.
4
.
K
.Y
.
Lam,
A
o mula
o he
angen bundleo
lag
mani olds
and
ela ed
mani olds,
T ans
.Ame
.Ma h
.Soc
.
213(1975)305-314
.
5
.
P
.
Lb le
and
L
.
Smi h,
Line
bundles
o e
amed
mani olds,
Ma h
.
Z
.
138(1974),
35-52
.
Ma hema isches
Ins i u
de
Geo g-Augus -Uni e si á
Bunsens
.
3-5
3400
GS ingen