On focal submanifolds of isoparametric hypersurfaces and Simons formula

0.16 MB·English

by  Qichao Li

ON FOCAL SUBMANIFOLDS OF ISOPARAMETRIC
HYPERSURFACES AND SIMONS FORMULA
QICHAOLI ANDLI ZHANG†
5
1
0
2
Abstract. Thefocalsubmanifoldsofisoparametric hypersurfacesinspheresareall

n
minimalWillmoresubmanifolds,mostlybeingA-manifoldsinthesenseofA.Graybut
a
J rarely Ricci-parallel ([QTY],[LY],[TY3]). Inthispaperwestudythegeometryofthe

8 focalsubmanifoldsviaSimonsformula. Weshowthatallthefocalsubmanifoldswith
2 g ≥ 3 are not normally flat by estimating the normal scalar curvatures. Moreover,

we give a complete classification of the semiparallel submanifolds among the focal
]
G submanifolds.
D
.
h
at 1. Introduction
m
[ Isoparametric hypersurfaces in a unit sphere are hypersurfaces with constant prin-

1 cipalcurvatures. Theyconsistofaone-parameterfamilyofparallelhypersurfaceswhich
v
laminates the unitspherewith two focal submanifolds at the end. It is remarkable that
3
4 the focal submanifolds of isoparametric hypersurfaces provide infinitely many spherical
0
submanifolds with abundant intrinsic and extrinsic geometric properties. For instance,
7
0 they are all minimal Willmore submanifolds in unit spheres and mostly A-manifolds in
.
1 the sense of A.Gray ([Gra]), except for two cases of g = 6 (cf. [QTY],[TY2],[LY],[Xie]).
0
As is well known, an Einstein manifold minimally immersed in a unit sphere is Will-
5
1 more, however the focal submanifolds are rarely Ricci-parallel, thus rarely Einstein (cf.
:
v [TY3]).
i
X
In this paper we study the geometry of the focal submanifolds in terms of Simons
r
a formula(see[Sim],[CdK]). Tostateourresultsclearly, wefirstlygivesomepreliminaries
on isoparametric hypersurfaces and their focal submanifolds in unit spheres.
AremarkableresultofMu¨nzner([Mu¨n])statesthatthenumberg ofdistinctprinci-
palcurvaturesmustbe1,2,3,4 or 6by thearguments of differential topology. Moreover,
the isoparametric hypersurfaces can be represented as the regular level sets in a unit
sphere of some Cartan-Mu¨nzner polynomial F. By a Cartan-Mu¨nzner polynomial, we
2000 Mathematics Subject Classification. 53A30, 53C42.
Key words and phrases. isoparametric hypersurface, focal submanifold, semiparallel, normally flat.
† The second authoris the corresponding author.
1
2 Q.C.LIANDL.ZHANG
meanahomogeneouspolynomialF of degreeg onRn+2 satisfyingtheso-called Cartan-
Mu¨nzner equations:
|∇F|2 = g2r2g 2, r = |x|,
−
( ∆F = crg−2, c = g2(m2−m1)/2,
where ∇F and ∆F are the gradient and Laplacian of F on Rn+2. The function
f = F|Sn+1 takes values in [−1,1]. The level sets f−1(t) (−1 < t < 1) gives the
isoparametric hypersurfaces. In fact, if we order the principal curvatures λ > ··· > λ
1 g
with multiplicities m ,··· ,m , then m = m (mod g), in particular, all multiplici-
1 g i i+2
ties are equal when g is odd. On the other hand, the critical sets M := f 1(1) and
+ −
M := f 1(−1) are connected submanifolds with codimensions m +1 and m +1 in
− 1 2
−
Sn+1, called focal submanifolds of this isoparametric family.
The isoparametric hypersurfaces Mn with g ≤ 3 in Sn+1 were classified by Cartan
to behomogeneous ([Car1],[Car2]). For g =6, Abresch ([Abr]) proved that m = m =
1 2
1 or 2. Dorfmeister-Neher ([DN]) and Miyaoka ([Miy3]) showed they are homogeneous,
respectively. For g = 4, all the isoparametric hypersurfaces are recently proved to
be OT-FKM type ([OT],[FKM]) or homogeneous with (m ,m ) = (2,2),(4,5), except
1 2
possibly for the unclassified case with (m ,m ) = (7,8) or (8,7) ([CCJ],[Imm],[Chi]).
1 2
Recently, TangandYan([TY3])providedacompletedeterminationforwhichfocal
submanifolds of g = 4 are Ricci-parallel except for the unclassified case (m ,m ) =
1 2
(7,8) or (8,7). More precisely, they proved
Theorem 1.1. ([TY3]) For the focal submanifolds of isoparametric hypersurfaces in
spheres with g = 4, we have
(i) TheM ofOT-FKMtypeisRicciparallel ifandonlyif(m ,m )= (2,1),(6,1),
+ 1 2
or it is diffeomorphic to Sp(2) in the homogeneous case with (m ,m ) = (4,3);
1 2
While the M of OT-FKM type is Ricci-parallel if and only if (m ,m ) =
1 2
−
(1,k).
(ii) For (m ,m ) = (2,2), the one diffeomorphic to G (R5) is Ricci-parallel, while
1 2 2
the other diffeomorphic to CP3 is not.
e
(iii) For (m ,m ) = (4,5), both are not Ricci-parallel.
1 2
Inspired by Tang and Yan’s result, we give a complete classification of the semi-
parallel submanifolds among all the focal submanifolds via Simons formula.
Recall that semiparallel submanifolds were introduced by Deprez in 1986 (see
[Dep]), as a generalization of parallel submanifolds. Given an isometric immersion
f : M → N, denote by B and ∇¯ its second fundamental form and the connection in
(tangent bundle)⊕(normal bundle), respectively, the immersion is said to be semipar-
allel if
(1) (R¯(X,Y)·B)(Z,W) := (∇¯2 B)(Z,W)−(∇¯2 B)(Z,W) = 0
XY YX
ON FOCAL SUBMANIFOLDS OF ISOPARAMETRIC HYPERSURFACES AND SIMONS FORMULA3
for any tangent vectors X,Y,Z and W to M, where R¯ is the curvature tensor of the
connection ∇¯ in (tangent bundle)⊕(normal bundle). An elegant survey on the study
of semiparallel submanifolds in a real space form can be found in [Lum].
Our main results state as follows:
Theorem 1.2. For focal submanifolds of isoparametric hypersurfaces in unit spheres
with g ≥ 3, we have
(i) For g = 3: all the focal submanifolds are semiparallel.
(ii) For g = 4: the M of OT-FKM type with (m ,m ) = (2,1),(6,1), the M of
+ 1 2 +
homogeneous OT-FKM type with (m ,m ) = (4,3), all of the M of OT-FKM
1 2
−
type with (m ,m ) = (1,k) and the focal submanifold diffeomorphic to G (R5)
1 2 2
with (m ,m )= (2,2) are semiparallel. None of the others are semiparallel.
1 2
e
(iii) For g = 6: none of the focal submanifolds are semiparallel.
Remark 1.1. Our results work on the unclassified case of g = 4, (m ,m ) = (7,8) or
1 2
(8,7).
This paper is organized as follows. In Section 2, we firstly state some prelimi-
naries on geometry of submanifolds and estimate the normal scalar curvatures of focal
submanifolds. In the next section, we proved Theorem 1.2 via Simons formula .
2. Notations and Preliminaries
Let f : Mn → Sn+p be a compact n-dimensional minimal submanifold of the
unit sphere. Denote by B the second fundamental form, ∇¯ the covariant derivative
with respect to the connection in (tangent bundle) ⊕ (normal bundle), ∇ (∇ ) the
⊥
induced Levi-Civita (normal) connection in the tangent (normal) bundle, respectively.
Choosing a local field ξ ,··· ,ξ of orthonormal frames of T M, we set A = A , the
1 p ⊥ α ξα
shape operator with respect to ξ .
α
The first covariant derivative of B is defined by
(∇¯ B)(Z,W) = (∇¯B)(X,Z,W) := ∇ B(Z,W)−B(∇ Z,W)−B(Z,∇ W).
X ⊥X X X
By the Codazzi equation, (∇¯ B)(Z,W) is symmetric in the tangent vectors X,Z and
X
W.
Define the second covariant derivative of B by
(∇¯2 B)(Z,W) := (∇¯ (∇¯B))(Y,Z,W) = (∇¯(∇¯B))(X,Y,Z,W),
XY X
where X,Y,Z,W ∈ TM. And the (rough) Laplacian of B is defined by
n
(∆B)(X,Y) := (∇¯2 B)(X,Y),
eiei
i=1
X
4 Q.C.LIANDL.ZHANG
where e ,··· ,e is a local orthonormal frame of TM.
1 n
The famous Simons formula states as follows:
Lemma 2.1. ([Sim] [CdK]) Denote by kBk2 the squared norm of the 2nd fundamental
form of a submanifold Mn minimally immersed in unit sphere Sn+p, then
1
(2) ∆kBk2 = k∇¯Bk2+hB,∆Bi
2
p p
= k∇¯Bk2+nkBk2− hA ,A i2− k[A ,A ]k2,
α β α β
α,β=1 α,β=1
X X
where hA ,A i := tr(A A ), k[A ,A ]k2 := −tr(A A −A A )2 and ∆ := tr(∇2),
α β α β α β α β β α
the Laplace operator of Mn.
The scalar curvature of the normal bundle is defined as ρ = kR k, where R is
⊥ ⊥ ⊥
the curvature tensor of the normal bundle. By the Ricci equation hR (X,Y)ξ,ηi =
⊥
h[A ,A ]X,Yi, the normal scalar curvature can be rewritten as
ξ η
p
ρ = k[A ,A ]k2.
⊥ α β
v
uα,β=1
u X
Clearly, the normal scalar curvaturetis always nonnegative and the Mn is normally flat
if andonly if ρ = 0, and by a resultof Cartan, which is equivalent to thesimultaneous
⊥
diagonalisation of all shape operators A .
ξ
Lemma 2.2. ([CR])The focal submanifolds of isoparametric hypersurfaces are aus-
tere submanifolds in unit spheres, and the shape operators of focal submanifolds are
isospectral, whose principal curvatures are cot(kπ), 1 ≤ k ≤ g −1 with multiplicities
g
m or m , alternately.
1 2
Proposition 2.1. For focal submanifolds of isoparametric hypersurfaces in Sn+1 with
g ≥ 3, the squared norms of the second fundamental forms kBk2 are constants:
(i) For the cases of g = 3 and multiplicities m = m := m,
1 2
2
kBk2 = m(m+1), m = 1,2,4,8.
M 3
±
(ii) For the cases of g = 4 and multiplicities (m ,m ),
1 2
kBk2 = 2m (m +1).
M+ 2 1
Similarly,
kBk2 = 2m (m +1).
M 1 2
−
(iii) For the cases of g = 6,
20
kBk2 = m(m+1), m = 1,2.
M 3
±
ON FOCAL SUBMANIFOLDS OF ISOPARAMETRIC HYPERSURFACES AND SIMONS FORMULA5
Proof. The proof follows easily from kBk2 = p tr(A2) and Lemma 2.2. (cid:3)
α=1 α
Theorem 2.1. For focal submanifolds of isPoparametric hypersurfaces in Sn+1 with
g ≥ 3, the normal scalar curvatures are estimated as follows:
(i) For the cases of g = 3 and multiplicities m = m =: m,
1 2
2
ρ = m 2(m+1), m ∈ {1,2,4,8}.
⊥M 3
±
p
(ii) For the cases of g = 4 and multiplicities (m ,m ),
1 2
ρ ≥ 2m m (m +1), ρ ≥ 2m m (m +1).
⊥M+ 1 2 1 ⊥M 1 2 2
−
(iii) For the cases opf g = 6 and multiplicities m p= m =: m,
1 2
8 80
(3) (ρ )2 = (72+ )m2(m+1), (ρ )2 = m2(m+1) m ∈{1,2}.
⊥M+ 9 ⊥M 9
−
As a result, all the focal submanifolds with g ≥ 3 are not normally flat.
Proof. By definition, the normal scalar curvature is given by
p p
(4) (ρ )2 = k[A ,A ]k2 = tr(A A −A A )(A A −A A )
⊥ α β α β β α β α α β
α,β=1 α,β=1
X X
p p
= 2tr (A2A2)−2tr (A A )2.
α β α β
α,β=1 α,β=1
X X
We estimate the normal scalar curvatures case by case.
(i) For the cases of g = 3 and multiplicities m = m =: m, n = 2m, p = m+1.
1 2
Let ξ be an arbitrary unit normal vector of a focal submanifold, by Lemma
2.2, the shape operator A has two distinct eigenvalues, 1 and − 1 , with the
ξ √3 √3
same multiplicities m. It follows easily that
1
(5) (A )2 = Id.
ξ
3
Since the shape operators defined on the unit normal bundle are isospectral,
for any two orthogonal unit normal vectors ξ and η, A still satisfies
1 (ξ+η)
√2
1
A2 = Id,
1 (ξ+η) 3
√2
that is
1 1 1 1 1
A2+ A A + A A + A2 = Id,
2 ξ 2 ξ η 2 η ξ 2 η 3
hence
A A +A A = 0,
ξ η η ξ
which follows
(6) tr(A A )2 = −tr(A2A2)= −tr(A2A2).
ξ η η ξ ξ η
6 Q.C.LIANDL.ZHANG
Substituting (5) and (6) into (4), one has
2m m(m+1) 8m2(m+1)
(ρ )2 = 8 tr(A2A2)= 8· · = .
⊥M α β 9 2 9
±
1 α<β p
≤X≤
(ii) For the cases of g = 4 and multiplicities (m ,m ): Recall a formula of Ozeki-
1 2
Takeuchi (see [OT] p.534 Lemma. 12. (ii))
A = A2A +A A2 +A A A , α 6= β.
α β α α β β α β
Multiply by A on both sides,
β
A A = A2A A +A A3 +A A A2, α 6= β.
α β β α β α β β α β
TakingtracesofbothsidesandobservingthatA3 = A forafocalsubmanifold
α α
with g = 4, we arrive at that
hA ,A i:= tr(A A )= 0, α 6= β.
α β α β
On the other hand, for a focal submanifold with g = 4, says M , kBk2 =
+
2m (m +1) (by Proposition 2.1), hA ,A i = 2m (by Lemma 2.2). Substi-
2 1 α α 2
tuting these into Simons’ formula (2), we conclude that
p
|[A ,A ]|2 = k∇¯Bk2+2m (m +1)(2m +m )−4m2(m +1)
α β M+ 2 1 2 1 2 1
α,β=1
X
= k∇¯Bk2+2m m (m +1) ≥ 2m m (m +1).
1 2 1 1 2 1
The calculations for M are completely similarly.
−
(iii) For the cases of g = 6 and multiplicities m = m =: m, all the focal subman-
1 2
ifolds are homogeneous, and the result follows from a straightforward calcula-
tion and the explicit expressions for A ’s from Miyaoka ([Miy1] [Miy2]).
α
(cid:3)
3. Semiparallel submanifolds and Simons formula
For a semiparallel submanifold Mn immersed in Sn+p, denote by e ,··· ,e a local
1 n
orthonormal frame of TM, X,Y ∈ TM, and H the mean curvature vector of Mn, then
n n
(7) (∆B)(X,Y) := (∇¯2 B)(X,Y)= (∇¯2 B)(e ,e )
eiei XY i i
i=1 i=1
X X
n
= ∇¯2 ( B(e ,e )) = ∇¯2 (H) = 0,
XY i i XY
i=1
X
where the second equality follows from the definition of semiparallel submanifolds and
Codazziequation,andthethirdfromthecommutativityofthecovariantdifferentiations
and taking trace. By the arbitrariness of X,Y ∈ TM, one concludes that ∆B = 0 for
a minimal semiparallel submanifold.
ON FOCAL SUBMANIFOLDS OF ISOPARAMETRIC HYPERSURFACES AND SIMONS FORMULA7
On the other hand, for a focal submanifold of an isoparametric hypersurface,
kBk2 = const. (by Proposition 2.1), H = 0 (by Lemma 2.2), combining with the Si-
mons formula 1∆kBk2 = k∇¯Bk2+hB,∆Bi one arrives at that k∇¯Bk2+hB,∆Bi= 0,
2
thus a focal submanifold of an isoparametric hypersurface is semiparallel if and only
if k∇¯Bk2 = 0, namely, it is a parallel submanifold. It remains to give a complete
classification of the focal submanifolds satisfying k∇¯Bk2 = 0.
It is convenient to define the covariant derivative (∇¯ A) of B along ξ ∈ U M
X α α ⊥
by h(∇¯ A) Y,Zi= h(∇¯B)(X,Y,Z),ξ i. It follows easily that
X α α
p
(∇¯ A) Y = (∇ A )Y − s (X)A (Y),
X α X α αβ β
β=1
X
p
where s is the normal connection defined by ∇ ξ = s (X)ξ . Obviously,
αβ ⊥X α β=1 αβ β
k∇¯Bk2 = p k(∇¯A) k2, thus
α=1 α P
(8) P k∇¯Bk2 = 0 ⇔ (∇¯ A) = 0 for all α = 1,··· ,p.
X α
The Ricci curvature tensor can be derived from the Gauss equation and the mini-
mality of the focal submanifolds as Ric = (n−1)Id− p A2. Moreover, by the fact
α=1 α
s +s = 0, one has
αβ βα P
p p p p
∇ Ric = − (∇ A) A − A (∇ A) =− (∇¯ A) A − A (∇¯ A) ,
X X α α α X α X α α α X α
α=1 α=1 α=1 α=1
X X X X
thus k∇¯Bk2 = 0 implies ∇Ric = 0 by (8), namely, Ricci-parallel. Putting all the facts
above together, we conclude that
Lemma 3.1. A focal submanifold of an isoparametric hypersurface is semiparallel if
and only if it is a parallel submanifold. In particular, a semiparallel focal submanifold
must be Ricci-parallel.
Proof of Theorem 1.2.
(1) For the cases g = 3, n = 2m, p = m+1, we have
p
2 8m2(m+1) 2m
kBk2 = m(m+1), k[A ,A ]k2 = , hA ,A i= δ .
α β α β αβ
3 9 3
α,β=1
X
Substituting these into Simons formula (2), one has
p p
k∇¯Bk2 = hA ,A i2+ k[A ,A ]k2−nkBk2
M α β α β
±
α,β=1 α,β=1
X X
(4m2)(m+1) 8m2(m+1) 4
= + − m2(m+1)
9 9 3
= 0,
thus all the focal submanifolds with g = 3 are semiparallel by Lemma 3.1.
8 Q.C.LIANDL.ZHANG
Remark 3.1. Each focal submanifold with g = 3 must be one of the Veronese em-
beddings of projective planes FP2 into S3m+1, where F is the division algebra R, C,
H, or O for m = 1,2,4, or 8, respectively. In facts, all the standard Veronese em-
beddings of compact symmetric spaces of rank one into unit spheres are known to be
parallel submanifolds [Sak]. Here we have provided a new proof for the cases of FP2,
F = R,C,H or O.
(2) For the cases g = 4 and multiplicities (m ,m ), for the focal submanifolds, say
1 2
M , n = 2m +m , p = m +1, we have
+ 2 1 1
(9) kBk2 = 2m (m +1), hA ,A i = 2m δ .
M+ 2 1 α β M+ 2 αβ
Recall a formula of Ozeki-Takeuchi (see [OT], p.534, Lemma. 12. (ii))
A = A2A +A A2 +A A A , α 6= β.
α β α α β β α β
Multiply by A and take traces on both sides,
α
tr(A2) = 2tr(A2A2)+tr(A A )2,
α α β α β
then
m kBk2 = m (m +1)tr(A2)= 2 tr(A2A2)+ tr(A A )2.
1 1 1 α α β α β
α=β α=β
X6 X6
On the other hand, by the formula (A )3 = A for 1 ≤α ≤ p (see [OT] p.534 Lemma.
α α
12. (i)),
3kBk2 = 2 tr(A2A2)+ tr(A A )2,
α β α β
α=β α=β
X X
thus
p p
(10) (m +3)kBk2 = 2 tr(A2A2)+ tr(A A )2.
1 α β α β
α,β=1 α,β=1
X X
Substituting (10) into (4), one concludes that for M
+
p p p
(11) k[A ,A ]k2 = 6tr( A2 A2)−2(m +3)kBk2.
α β M+ α β 1
α,β=1 α=1 β=1
X X X
Similarly, for M ,
−
p p p
k[A ,A ]k2 = 6tr( A2 A2)−2(m +3)kBk2.
α β M α β 2
−
α,β=1 α=1 β=1
X X X
ON FOCAL SUBMANIFOLDS OF ISOPARAMETRIC HYPERSURFACES AND SIMONS FORMULA9
Moreover, substituting (9) and (11) into (2), one arrives at
p p
k∇¯Bk2 = 4m2(m +1)+6tr( A2 A2)−2(m +3)kBk2
M+ 2 1 α β 1
α=1 β=1
X X
(12) −(2m +m )kBk2
2 1
p p
= 6tr( A2 A2)−6m (m +1)(m +2).
α β 2 1 1
α=1 β=1
X X
Similarly,
p p
(13) k∇¯Bk2 = 6tr( A2 A2)−6m (m +1)(m +2).
M α β 1 2 2
−
α=1 β=1
X X
We firstly show that the focal submanifolds of the cases g = 4, (m ,m ) = (7,8) or
1 2
(8,7) can not be semiparallel. In fact, by the inequality tr(A2) ≥ (trA)2 for symmetric
n
matrices,
p
6
k∇¯Bk2 ≥ (tr A2)2−6m (m +1)(m +2)
M+ m +2m α 2 1 1
1 2
α=1
X
24
= m2(m +1)2 −6m (m +1)(m +2)
m +2m 2 1 2 1 1
1 2
6m m (m +1)
1 2 1
= (2m −m −2),
2 1
m +2m
1 2
Clearly, in the cases of (m ,m ) = (7,8) or (8,7), one has k∇¯Bk2 > 0, k∇¯Bk2 > 0.
1 2 M+ M
−
By Lemma 3.1, the focal submanifolds of the cases g = 4, (m ,m ) = (7,8) or (8,7)
1 2
are not semiparallel.
We nextly show the Ricci-parallel focal submanifolds listed in Theorem 1.1 are all
semiparallel case by case.
For the M of OT-FKM type with (m ,m ) = (1,k), which can be characterized
1 2
−
as M = {(xcosθ,xsinθ)|x ∈Sk+1(1),θ ∈ S1(1)} = S1(1)×Sk+1(1)/(θ,x) ∼ (θ+π,−x)
−
(see [TY1] Proposition 1.1). Clearly, the Ricci tensor of M can be characterized as
Ric = 0⊕k·Id . However, by the Gauss equation R−ic = (k+1)Id− k+1 A2,
(k+1) (k+1) α=1 α
×
one has
P
k+1
(14) A2 = (k+1)⊕Id .
α (k+1) (k+1)
×
α=1
X
Substituting (14) into (13), one arrives at
k∇¯Bk2 = 6((k+1)2+(k+1))−6(k+1)(k+2) = 0.
Mk+2
−
According to Lemma 3.1, M ’s of OT-FKM type with (m ,m ) = (1,k) are semi-
1 2
−
parallel submanifolds. Furthermore, the isoparametric families of OT-FKM type with
10 Q.C.LIANDL.ZHANG
multiplicities (2,1), (6,1) are congruent to those with multiplicities (1,2), (1,6), re-
spectively ([FKM]), thus M ’s of OT-FKM type with (m ,m ) = (2,1) or (6,1) are
+ 1 2
also semiparallel submanifolds.
The M of OT-FKM type with (m ,m ) = (4,3) (the homogeneous case), as
+ 1 2
proved in ([QTY]), is an Einstein manifold. In this way,
p
kBk2
(15) A2 = ·Id = 3·Id .
α 10 10×10 10×10
α=1
X
Substituting (15) into (12), one arrives at
k∇¯Bk2 = 6·9·10−6·3·5·6 = 0.
M10
+
According to Lemma 3.1, M of OT-FKM type with (m ,m ) = (4,3) (the homoge-
+ 1 2
neous family) is semiparallel submanifold.
The focal submanifold diffeomorphic to G (R5) in the exceptional homogeneous
2
case with (m ,m )= (2,2), as proved in ([QTY]), is an Einstein manifold. In this way,
1 2
e
p
kBk2
(16) A2 = ·Id = 2·Id .
α 6 6×6 6×6
α=1
X
Substituting (16) into (12), one arrives at
k∇¯Bk2 = 6·4·6−6·2·3·4 = 0.
M10
According to Lemma 3.1, the focal submanifold diffeomorphic to G (R5) in the excep-
2
tional homogeneous case with (m ,m )= (2,2) is semiparallel submanifold.
1 2
e
The proof of the second part of Theorem 1.2 is now complete.
(3) For the cases g = 6, n = 5m,p = m+1, we have
20 20m
kBk2 = m(m+1), hA ,A i= δ .
α β αβ
3 3
Substituting these and (3) into Simons formula (2), one has
p p
k∇¯Bk2 = hA ,A i2+ k[A ,A ]k2−nkBk2
M+ α β α β
α,β=1 α,β=1
X X
(400m2)(m+1) 8 100
= +(72+ )m2(m+1)− m2(m+1)
9 9 3
= 84m2(m+1) > 0.
Similarly,
k∇¯Bk2 = 20m2(m+1) > 0.
M
−
Hence, none of the focal submanifolds with g = 6 are semiparallel by Lemma 3.1.
The proof of Theorem 1.2 is now complete. (cid:3)