its corresponding normal space. A totally real immersion is called. Lagrangian if dimR. M=dimC. M. For Lagrangian immersions in complex. Euclidean n-space.
Japan. J. Math. Vol. 26, No. 1, 2000
Some
new
obstructions
Lagrangian
isometric
to minimal
and
immersions
Dedicated to Professor Tadashi Nagano on the occasion of his seventieth birthday By
Bang-Yen
CHEN
(Received September 24, 1997) (Revised January 7, 1999) (from Kyushu Journal of Mathematics)
1.
Introduction
The main purpose of this paper is to introduce a new type of Riemannian curvature invariants and to show that these new invariants have interesting appli cations to severalareas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions. One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility)of a Riemannian manifold in a Euclidean space (or more generally, in a space form). According to a well-knowntheorem of J.F. Nash, every Riemannian manifoldcan be isometrically immersed in some Euclidean spaces with sufficientlyhigh codimension. In order to study this fundamental problem, in view of Nash's theorem, it is natural to impose a suitable condition on the immersions. For instance, if one imposes the minimality condition on the immersions, it leads to PROBLEM 1. Given a Riemannian manifold M, what are the necessary con ditions for M to admit a minimal isometric immersion in a Euclidean m-space Em? It is well-knownthat for a minimal submanifold in Em, the Ricci tensor satis fies Ric_??_0.For many years this was the only known general necessaryRiemannian condition for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Themainresultsof this articlewerepresentedat the 3rd PacificRimGeometryConference heldat Seoul,Koreain December 1996;alsopresentedat the 922ndAMSmeetingheldat Detroit, Michiganin May1997.
106
BANG-YEN CHEN
An is
immersion
called
structure
J
space.
A
mov
real
of
real
[12]
states
of to
tangent
is
in
M
is
Cn
of
called
only
is
point
of
Hermitian if
its
almost
implies
normal M.
Cn,
a result
a Lagrangian
of
Gro
immersion TM_??_C
that
compact
M
complex
M=dimC
n-space
complexification
result
manifold
the
corresponding
if dimR
admits
if the
for
a
into
Euclidean M
Gromov's
a 3-manifold
M
Lagrangian
complex
immersions
Riemannian
of
n-manifold
trivial.
in
geometry)
space
in
if and
M
symplectic
tangent
a compact
Lagrangian
bundle From
each
immersion
that
bundle
n-manifold in
immersions
isometric)
obstruction
Riemannian isotropic
maps
totally Lagrangian
a
(or
M
For
necessary gent
of
totally
there
is
3-manifolds
in
(not
of no
the
tan
topological
C3,
because
the
trivial.
views,
it is
natural
to
ask
the
following
basic
ques
tion. PROBLEM manifold The A
class
when
do
author
in
exist
slant
slant
3.
to
The ƒÂ . He
admit
author
squared
mean
applying
his
been
(cf.
In
n1,
nk
...,
is
are
[3]
the
for
arbitrary
he
was
extensively paper
we
Riemannian are
extend
the
invariants
integers
satisfying
ask
a
satisfying
the few
invariant ƒÂ denoted
On
is
7].
immersions:
real
real.
[2,
admits
no
(or
A no
slant
Lagrangian
slant
the
for
a
immersions
other
hand,
there
in some equality
years a
this
space
solutions
for
denoted
invariant
Riemannian
to of
the
forms.
[3,
By
also
2
have
4, 8-11]).
n-manifold
We
the
1,
inequality
to
and ƒÂ(n1,...,nk),
2_??_n1_??_..._??_nk0
and
nature.
a personal
communication
(dated
April
(2.5) for
Riemannian
nian
space
forms.
3.
Sharp
We
recall
LEMMA
Then
n-manifolds
inequalities the
3.1.
2a1a2_??_ā,
following Let
with
the
equality
holding
only
involving ƒÂ(n1,...,nk) algebraic
a1,...,an,ā
equality
and
lemma be
holding
n+1
if
from real
and
only
[3]. numbers
if
such
a1+a2=a2=...=an.
that
for
Rieman
Obstructions
For positive
each
(n1,
to
...,
constants
minimal
and
Lagrangian
let
c(n1,
nk)•¸S(n),
given
isometric
...,
nk)
immersions
and
b(n1,
one
for
109
...,
nk)
denote
the
by
(3.1)
(3.2) The
following
theorem
THEOREM space
form
3.2.
is
Given
Rm(•¸)
of
the
an
constant
most
fundamental
n-dimensional sectional
submanifold
curvature •¸,
we
this
paper.
M
in
a
p •¸
M
if
Riemannian
have
(3.3) for
any
k-tuple
The there M
exists in
(n1,
equality an
Rm(•¸)
...,
nk)•¸S(n).
case
of
inequality
the
following
orthonormal
at p
take
basis
(3.3) e1,
...,
holds em
at
at p,
a point such
that
the
shape
and
only
if,
operators
of
forms:
(3.4)
where
I
is
an
identity
matrix
and
each
Arj
is
a
symmetric
nj•~nj
submatrix
that
(3.5) PROOF. Let
M
curvature
If be tensor
k=0,
this
a submanifold R
of
M
was of
satisfies
done
in
[5].
a Riemannian
Hence,
we
space
the
equation
of
and the squared
mean
curvature
assume
form
Rm(•¸).
k_??_1. The
Gauss:
(3.6)
The
scalar
(3.7)
curvature
H2 of M satisfy
Riemannian
such
110
BANG-YEN
where •ah•a2 Let
is
the
squared
(n1,...,nk)•¸S(n).
norm
of
the
CHEN
second
fundamental
form
h
of
M
in
Rm(•¸).
Put
(3.8) Then from (3.7) and (3.8) we find (3.9) Let L1, ..., Lk be mutually orthogonal subspaces of TpM with dim Lj=nj, j=1,...,k.
By choosing an orthonormal basis e1, ..., em at p such that
and en+1 is in the direction of the mean curvature vector, we obtain from (3.9) that (3.10) where
ai=hn+1ii,
i=1,...,n
and 'y = n + k
We put
Equation (3.10) is equivalent to (3.11)
where (3.12)
n3.
Obstructions
to minimal
and Lagrangian
isometric
immersions
111
Application of Lemma 3.1 to (3.11) yields (3.13)
On the other hand, (2.2) and Gauss' equation yield (3.14)
Combining (3.13) and (3.14), we get
(3.15)
where
0=Z (2.3),
If
equality
(3.15)
the are
(3.13),
Since the
exact
THEOREM
proof
of
proof
of
3.3.
any
k-tuple
p.
Let
quaternionic)
(n1,...,nk)•¸S(n).
a
by 3.2
be
Xpk).
(3.3).
point
p,
this
case,
(3.4)
verified
M
1)U...U(p
obtain
In
obtain
Theorem
(respectively,
(respectively,
at
Theorem
(3.16) for
at we
be
we
holds
equalities
can
xz
(3.15),
(3.3)
(3.15),
converse the
and in
and
Kaehlerian phic
,02=(z
(3.8)
actually
(3.14) The
tion,
U...Ut
From
and
then by
the
applying
based
also
an
n-dimensional
quaternionic) sectional
yields
space
Lemma
(3.13)
and
3.1,
(3.11),
Gauss'
equa
computation.
only
3.2
in
(3.5).
a straight-forward is
inequalities
on the
Lemma
curvature
and
following. totally
form
3.1
real
Mm(4•¸) 4•¸.
Then
of we
submanifold constant have
of holomor
a
112
BANG-YEN
4.
Some
Theorem
applications 3.2
THEOREM
gives
Let
admits
at
a
In
for
some
follows: totally
the
each
integer
vanishes immersion.
--*
Emj, of
_??_ 2 with
4.2.
scalar
If
any
Let
then
j
in =
and
to
there
Problem
exists
1.
a k-tuple
Riemannian
space
form
fri
the
the
>
be every
no
minimal
iso
(ni,...,n,)•¸S(n),
sharp.
This
minimal
can
be
submanifolds
n3)-space
product
Mn11,...,Mnkk Then
k
is
Riemannian
Clearly,
curvatures_??_0.
be
admits
codimension. k-tuple
4.1
, k,
a Euclidean
M of
each
Theorem
1, ...
of
identically.
COROLLARY
in
regardless
n_??_2
given
immersion
minimal
solution
that
immersion
space
invariant ƒÂ(n1,...,nk)
En-‡”nj
new
n-manifold.
such
Euclidean
For
:
geodesic
Then
following
(n1,...,nk)•¸S(n),
any
4.1.
f
p•¸M
isometric
on ƒÂ(n1,...,nk)
Let
the
a Riemannian
a point
k-tuple in
condition
to
if
immersion REMARK
the
be
minimal
particular,
point
metric
no
immediately M
and
M
Rm(•¸).
rise
4.1.
(ni,...,n,)•¸S(n)
then
CHEN
into
a Euclidean Mn11•~...•~Mnkk•~
immersion
f1•~...fk•~ƒÇ
manifolds
minimal
isometric
as
of
a
space.
product
Riemannian
seen and Ă
is
a
dimensions
immersion
(4.1) of
Mil
fi
xx
x
...
x
f k x
totally
scalar
vanishes
of
Since
identically.
such
sign way
of
k
En-E
the
ni
in
minimal
any
Euclidean
immersions
Mn11,...,Mnkk
curvatures_??_0,
equality
Mnjj
k x
m-space
f:
MJ
is
'
Emj,
a product j=
immersion 1,...,
k
and
a
immersion Ă.
PROOF.
in
t
geodesic
with
the
M~
the
Thus, of
that Riemannian
for
inequality enl+•••+nj+i, product,
are
Riemannian
invariant ƒÂ(n1,...,nk) any
minimal
(3.3)
holds
then
of Ml
immersion identically.
• •. ,
are the
second
manifolds
f
of 1
of Mn
Hence, tangent fundamental
k x E'E
in a Euclidean if we
to
dimensions_??_2
• x M~
the form
choose j-th h
space, e1,...,en
component of
f
satisfies
(4.2) for any Xj tangent to the j-th component of the Riemannian product according to Theorem 3.2. Now, Corollary 4.2 followsfrom Moore's lemma [14].
Obstructions REMARK 4.2.
which
satisfy The
Similar
result
holds
for isometric
immersions
113
immersions
H2=b(n1,...,nk)/c(n1,...,nk).
following
result
THEOREM damental
to minimal and Lagrangian isometric
4.3.
provides
Let
group ƒÎ1
or
(n1,...,nk)•¸S(n)
M
with
in
any
Lagrangian
isometric
be null
such
immersion
new a
solutions
compact
first
to
Riemannian
betti
n-torus
i.e., then
CTn
immersion
in
or
any
in
2 and
3.
n-manifold
number,
that ƒÂ(n1,...,nk)>0,
complex
Problems
M
Cn.
complex
with
b1=0.
If
admits
In
no
or
is
isometric
M
in
fun
a k-tuple
slant
particular,
n-torus
finite
there
admits
complex
no
Euclidean
n-space. PROOF. ƒ¿
=0,
If
then
this
M
case,
H is
nowhere
zero.
is
cannot
that not
the
class be
the
3.2
1-form
and ƒ¦
is
on
a
implies
that
defined
[ƒ¦] group
is
R)
is
assume ƒ¿•‚0. for
immersion
the
mean
[2]).
in
a
curvature
Thus, ƒ¦
compact
a non-trivial
H1(M;
a-slant
(cf. is
many
In
flat
vector
by ƒ¦(X)=n(cscƒ¿)•qH,JX•r.
1-form M
with
submanifold.
with ƒÂ(n1,...,nk)>0 a
M
closed
manifold
a minimal
we
admits
Because
Therefore,
cohomology
M
Theorem
[ƒ¦]•¸H1(M;R).
exact.
first
be
zero
Kaehlerian is
n-manifold
If
Let ƒ¦
it
Hence,
Riemannian
then
a flat
thus
a contradiction.
compact
M,
of and
and ƒ¦
cohomology non-trivial.
represents is
class Hence,
nowhere which
b1•‚0
a zero,
implies and
M
is
simply-connected.
with
covering
suppose finite map
M
is
lifts
Riemannian
universal
covering
Applying
the
a-slant
is
Theorem
COROLLARY damental
group ƒÎ1
some
mersion
damental
k-tuple in
also
4.4. or
compact.
we
of Let
with
may
complex
If
COROLLARY
4.5. or
Let with
M b1=0.
n-space be
to is
the
a
universal
a-slant
immersion
compact the
Riemannian
and ƒÎ1
covering
for is
map
finite,
preserves
the the the
a contradiction.
the
following
two
a compact If
Riemannian
6(ni,..., then
projective
with ƒÂ(n1,...,nk)>0
case,
results
by
using
some
4.3.
be a compact
b1=0.
M M
yields
prove
Theorem
M
this
Because
case
(n1,...,nk)•¸S(n),
the
group ƒÎ1
Since
previous
3.3, that
of
space.
the
to
n-manifold In
immersion
covering space
similar
Riemannian
group ƒÎ1•‚0.
invariant ƒÂ(n1,...,nk),
arguments
a compact
fundamental
universal
for
submanifold
submanifold
yields
a
nowhere
Now, and
is
n-manifold
cohomology ƒ¦
a-slant
(n1,...,nk)•¸S(n).
Kaehlerian
Then ƒ¦
an
3.2
M
k-tuple
is
a complex
Theorem
Suppose some
M is
o(ni,...
n-manifold
nk)>2 M
admits
(n(n no
with
-1)
finite n3(nj
Lagrangian
isometric
fun -1)) im
CPn(4). Riemannian , nk)>2(~
n-manifold n3 (n3
with -1)
finite n(n
fun -1))
114
for
BANG-YEN
some
k-tuple
mersion
in REMARK
laries
4.4
(n1,...,nk)•¸S(n),
the
complex 4.3.
and
w:Sn•¨Cn
are
then
hyperbolic The
4.5
M
n-space
conditions
sharp.
defined
CHEN
For
admits
no
Lagrangian
isometric
im
CHn(-4).
on ƒÂ(n1,...,nk) example,
given
consider
in
Whitney's
Theorem
4.3,
Lagrangian
Corol
immersion
by
(4.3) with
y20+y21+...+y2n=1.
point
The
Whitney's
immersion
has
a unique
self-intersection
w(-1,0,...,0)=w(1,0,...,0). For
any
respect
to
n-tuple
the
(n1,...,nk)•¸S(n),
induced
metric;
we
have ƒÂ(n1,...,nk)_??_0
and ƒÂ(n1,...,nk)=0
only
on
at
the
Sn
unique
with
point
of
self-intersection. REMARK ometric curvature infra
of
face
in
Let and
p)
M =: 1•~Sn-1
each
must
let Ă:
=
F(s) 51
x to
4.3
5. In
A general other
MAXIMUM clidean
by
Lagrangian
submanifold
M
over
all
at unit
some
®
t(p),
Sn-1 a point
p
e
into Cn in Cn
x
Sn^1.
m-space
there
does
not
have
If
a k-tuple
(5.2)
(n1,...,nk)•¸S(n),
is
of
Cn M.
an
and
that
both
the
and
ideal
For
satisfy
Lagrangian sur
M. plane
given in
complex
isometric pair
must
Ricci
Lagrangian
hypersphere the
ge
the
by
En
at
defined
by
Lagrangian
immersion
{(u,p),(-u,-p)}
of and on
conditions
centered
extensor
b1(M)=1,
F(s)=
points
in S
moreover, M
on ƒÎ1(M)
is
a
and
for
positive b1(M)
in
removed.
PRINCIPLE. Em.
unit
each
in
compact
complex
Cn
f
carries
principle
we
the
. Clearly, ƒÎ1(M)=Z
shows be
maximum
hand,
the
--~
Then
which
on
following
namely,
the ƒÂ-invariant ƒÂ(n1,...,nk)
example cannot
Sn-1
in
every
is the
Cn;
vectors of
points
circle be
: 51
tangent
curvature
exist the Let
it
a direct
immersions
relationship
between
the
new
invariants.
following.
satisfies
M the
(5.1) for
unit
Gaussian
the
f
4.3 in
nonpositive be
Theorem
submanifolds
(n1,...,nk)•¸S(n), This
Theorem
runs the
of
Lagrangian
Sn-1•¨En(n_??_3)
k-tuple
the
be
Denote
constant.
On
u that
S1•¨C
origin.
f (s, of
where
F:
consequence
compact
means
C2
immediate
compact
every
this
M
the
An on
Ric(u)_??_0,
surfaces,
eis
4.4.
condition
then
be
an
equality
n-dimensional case
of
submanifold (3.3),
i.e.,
it
of satisfies
a
Eu
Obstructions
for
any
to minimal
(m1,...,mj)•¸S(n),
and Lagrangian
isometric
immersions
115
where
(5.3) PROOF. For Em,
It
an
follows
from
isometric
Theorem
3.2
Theorem
immersion
x:
3.2
with •¸=0.
M•¨Em
of
a
Riemannian
n-manifold
M
in
yields
(5.4) where Ģ0
is
the
Riemannian
invariant
on
M
defined
by
(5.5) Inequality
(5.4)
enables
us
to
introduce
the
notion
of
ideal
immersions
as
follows. DEFINITION Em
is
called
5.1. an
REMARK an
isometric
the
least
at
point
curvature a
5.1
on
Riemannian
the
each
an
k-tuple
on
receives
REMARK
5.2.
is
the
from
The
isometric
In
this
Laplacian Ģ
we
above
the
it
is
principle
an
yields
M
surrounding
thus
space the
mean
the
of
squared
amount
mean of tension
point. the
Laplacian
following
important
(5.1)
and between
the
that
receives
immersion
the that
fact M
that
isometric
in
for
a
given
automatically.
relationship and
the
equality
immersion
of
close
manifold
at
satisfies ideal
The
fact
an
M
identically.
that
measures
space
eigenvalues some
for
manifold;
M•¨Em
(5.4)
means
well-known
field
surrounding
x:
from
simply
maximum
establish
Riemannian
and
tension
of
Immersions).
by Ģ0(p))
(5.4)
n-manifold
case
immersion
Riemannian
the
Riemannian
equality
Ideal
ideal
submanifold
between
a
to the
then
section on
due
immersion
Relations
an
(given
another
(n1,...,nk)•¸S(n),
6.
tension
exactly in
point
submanifold
If
This is
manifold at
fact:
M. field
is
a
the of
M•¨Em of
of
satisfies
Interpretation
x:
amount
p
immersion if it
(Physical
vector
curvature
isometric
immersion
immersion possible
each
An
ideal
new
the
invariants
eigenvalues
invariants.
In
of
the
particular,
we improve a well-knownresult of T. Nagano obtained in [15]. In order to do so, we recall (cf.
[6]
for
Let M.
the notions
Denote
of order
and type
from the theory
of finite
type
submanifolds
details). M
be
a compact
by ăj
the
Riemannian j-th
nonzero
n-manifold eigenvalue
and Ģ of
the
the
Laplacian Ģ
Laplacian on
operator M.
of
116
BANG-YEN
For
an
isometric
immersion
x:
CHEN
M•¨Em
of
M
in
Em,
let
(6.1) denote
the spectral
decomposition
of x, where
x0 is center
of mass
of M in Em.
The set
(6.2) is called the order of the submanifold. The smallest element p in T(x) is called the lowest order of x and the supremum q of T(x) is called the highest order of x. The immersion is said to be of finite type if the highest order q is finite; and it is said to be of infinite type if the highest order q is infinite. Moreover, the immersion is said to be of k-type if T(x) contains exactly k elements. Clearly, the immersion is of 1-type if and only if p=q. In this case, the immersion is called a 1-type immersion of order {p}. For an isometric immersion of a compact Riemannian manifold M in a Eu clidean space, one has (cf. [6]) (6.3) where vol(M) denotes the volume of M and p, q are the highest order and the lowest order of M, respectively. Either equality sign of (6.3) holds if and only if the immersion is of 1-type with order {q} or order {p}, respectively. Since p_??_1,(6.3) implies immediately the followingresult [17]: (6.4)
THEOREM Riemannian
6.1.
Let
n-manifold.
x:
M•¨Em
be
an
isometric
immersion
of
a
compact
Then
(6.5) for
any
k-tuple
The of
order
(n1,...,nk)•¸S(n),
equality {q}
satisfying PROOF.
case
associated
of with
H2=Ģ0(n1,...,nk) Follows
where (6.5)
holds
if
(n1,...,nk),
q and
is
the
only
i.e.,
x
is
and
(6.3).
identically. from
Theorem
3.2
highest if
x
is
a 1-type
order a
of
1-type immersion
the
immersion.
ideal
immersion of
order
{q}
Obstructionsto minimal and Lagrangianisometricimmersions
117
Theorem 6.1 implies immediately the following. THEOREM6.2. If a compact Riemannian n-manifold M admits a 1-type iso metric immersion in a Euclidean space, then
(6.6) for
any
k-tuple
The if
the
of
an
in
terms
the
(n1,...,nk)•¸S(n),
equality
case
1-type
immersion
REMARK
6.1.
isometric
where
of
(6.6) is
of
our
an
ideal
Theorem
immersion new
Riemannian
holds
6.1 of
of
provides
is
the
order
k-tuple
associated us
a
1-type
immersion. if
to
and
only
(n1,...,nk).
estimate
the
manifold let
with
the
with
way
example,
2-spheres
of
(n1,...,nk)•¸S(n)
Riemannian
For two
a
immersion
a compact
invariants.
product
p
for
in
highest
order
a Euclidean
space
M=S2(1/a2)•~S2(1/b2)
curvatures
1/a2,
be 1/b2,
respectively.
Assume
for of
some M
and
natural
are
number
given
by
s•¸N0.
2/a2,
4Ģ0=1/(a2b2)>ăs,
isometric
q of
must
be The
of
any
first
of
isometric
the
s nonzero
6.1
implies in
of
Since Ģ0=1/(4a2b2)
that
any
codimension,
immersion
eigenvalues ă1, ă2,...,ăs
respectively.
S2(1/a2)•~S2(1/b2)
independent
order
the
Theorem
immersion
Consequently,
Then
6/a2,...,s(s+1)/a2,
the
highest
Euclidean
if b is
order
space
a small
S2(1/a2)•~S2(1/b2)
is
number, in
any
at
of
least the
any s+1.
highest
Euclidean
space
large. following
result
provides
a
sharp
relationship
between ă1
and
our
new
invariants. THEOREM irreducible M
isotropy
6.3.
If
M
action,
is then
a compact the
homogeneous
first
nonzero
Riemannian eigenvalue ă1
n-manifold of
the
Laplacian
with on
satisfies
(6.7) for
any
k-tuple
(n1,...,nk)•¸S(n).
Therefore,
we
have
(6.8) The equality sign of (6.8) holds if and only if M admits a 1-type ideal immer sion in a Euclidean space.
118
BANG-YEN
PROOF. neous ric
Follows
Riemannian immersion
of
Riemannian
order
REMARK [15],
In
general,
we
Theorem
and
inequality
6.4. M
k-tuple
yields
the to
Let
admits
where ƒÏ and ƒ¢0>ƒÏ
manifold
THEOREM
M an
(n1,...,nk),
the
fact
isotropy
that
every
action
compact
admits
of
homoge
a 1-type
a compact
isomet
homogeneous
constants.
have ƒ¢0_??_ƒÏ, also
and
invariants ƒÂ(n1,...,nk)
When k=0,
3.2
6.2
irreducible
namely, ƒÉ1_??_nƒÏ,
Riemannian
If
are
6.2.
Nagano
(1)
Theorem with
{1}
manifold
T.
pact
from
manifold
CHEN
ideal
for
a
ideal compact
the
reduces
to
a well-known
normalized
most
following
admit be
(6.7) is
scalar
Riemannian
necessary
result
curvature
of
of
M.
manifolds.
intrinsic
condition
for
a
com
immersions. Riemannian
immersion
in
n-manifold.
a Euclidean
space
associated
with
a
then
(6.9)
(2) If M satisfiesap
1),
(n and
>
A1=
admits
M
nL1(ni,...
8(n =
6.6
RPn(1); +
L~(2,
an
L 0
the
only
=
only
in
a
n-dimensional
if and
the =
following
4(n+
1),
o(n,
n,
2, 2, 2, 2, 2, 2, 2)
is
scalar
and Al
1),
immersion
positive
, nk)
=
CP1
ideal
For
for
Ao
6.8.
with
Theorem
=1
Al
48,
an
PROOF. ifold
Leo
1);
COROLLARY which
from
1),
n, n)
irreducible
0(n, +
for
have
n)
=
3)/n
(n+3)/n
for
HP(4)
0P2(4).
Hermitian
symmetric
arbitrary
homogeneous we
(n for
space
compact
=
=
=145/56
Euclidean
curvature,
facts: zo
Einstein
nă1=4Ą
space
codimension. Kaehler
[16].
man
Thus,
ai>
if
(6.12) The
coefficient
three
cases
of ƒÂ(n1,...,nk) occurs:
n1+n2=n.
In
compact
type,
irreducible
the
namely,
7.
equality
we case
EXAMPLE is
M
a
7.1.
a
is an
is
equal
(2)
k=1,
compact
to
type
with
some
imply
examples
integer
k•¸{0,1,...,[n/2]},
and
only
that
if
and
one (3)
symmetric geometric
(6.12)
of
the
k=2, space
of
properties
holds
unless
of n=2,
line.
satisfying
simple
if
Hermitian
projective
submanifolds
two n=n1+1,
irreducible
together
complex
some
inequality
hypersurface
of
is
of
provide of
M
observations
spaces
Examples
Here,
when
above
unless
(6.12)
k=0, n=2,
particular,
symmetric
k=0;
Sn-k
(1)
in
basic of
equality
submanifolds
which
satisfy
the
(3.3).
For
any of
En+1
satisfying
(5.1)
a spherical with
(n1,...,nk)=(2,...,2).
cylinder
Ek•~
120
BANG-YEN
EXAMPLE in
Rm(•¸)
lar,
a
7.2.
For
satisfies
the
horosphere
each
equality
in
a
in
(3.3)
EXAMPLE round
a2+b2=1,
is an
ideal
with
7.3.
k-sphere
with
in
(3.3)
n=2k, with
space
A
equality
number
hyperbolic
(n1,...,nk)=(2,...,2). the
even
CHEN
totally
a totally
umbilical
the
submanifold
be
curvature
product
embedding
embedding
In
satisfies
equality of
in
Rm(•¸)
particu (3.3)
also
with
satisfies
k=0.
constant
the
submanifold
(n1,...,nk)=(2,...,2).
H2k+1
Let ƒÇa:Sk(1/a2)•¨Ek+1
with
umbilical
in
E2k+2
which
1/a2.
For
satisfies
the
the
standard
any
two
equality
embedding
positive
in
(3.3)
numbers
with
of
a
a,
b
(n1,...,nk)
=(2,...,2). In some
general,
if l
integers
and
n-l
k>1,t,s,
Sl(1/a2)•~Sn-l(1/b2)
in
(n1,...,nk),
where
EXAMPLE ative
For
NB2k
austere
lies
(3.3)
with
in
M to
be
THEOREM
(8.1)
an
ideal
such
that l=kt, of
embedding
associated
austere we
the
n-l=ks
Riemannian
for product
with
the
k-tuple
and open
the
dense
with
submanifold
=1
and
identity
Sn+1(1)
(e,TPB2JC)
map
subset
relative
of
with
zero
rel
put
I (,)
inequalities
a submanifold we
put
U
nullity
Mm(4•¸)
for
from
of
n-2k.
=
NB2k
NB2k.
0}.
into
Moreover, NB2k
of
of JX, of result
Mm(4•¸) 8.1. with •¸