Topological obstructions for submanifolds in low codimension

0.24 MB·

by  Christos-Raent Onti

Topological obstructions for submanifolds in low
codimension
7
1
Christos-Raent Onti and Theodoros Vlachos
0
2

n
a
J Abstract

8
We prove integral curvature bounds in terms of the Betti numbers for compact
1
submanifolds of the Euclidean space with low codimension. As an application, we

] obtain topological obstructions for δ-pinched immersions. Furthermore, we obtain
G
intrinsic obstructions for minimal submanifolds in spheres with pinched second fun-
D
damental form.
.
h
t
a
m 1 Introduction
[

1 By the Nash’s embedding theorem, every Riemannian manifold can be isometrically im-
v mersed into a Euclidean space with sufficiently high codimension. On the other hand,
5
there are results that impose restrictions on isometric immersions with low codimension
2
0 (cf. [10,18–21,23]). Most of these obstructions are pointwise conditions on the range of
5
curvature. Here, we investigate obstructions for immersions with low codimension that
0
. involve total curvature. In particular, we are interested in the Ln/2-norm of the (0,4)-
1
0 tensor R scal/n(n 1) R , where R and scal denote the (0,4)-curvature tensor and
1
7 the scalar−curvature of−the induced metric g respectively, and R = (1/2)g 7g, where 7
1 (cid:0) (cid:1) 1
: stands for the Kulkarni-Nomizu product. Shiohama and Xu [25] gave a lower bound in
v
i terms of the Betti numbers for compact hypersurfaces in the Euclidean space Rn+1. For
X
higher codimension, they raised the following
r
a
Problem. Let Mn,n 3, be a compact n-dimensional Riemannian manifold which
≥
admits an isometric immersion into R2n−1. Does there exist a constant ε(n), depending
only on n, such that if
scal n/2
R R dM < ε(n)
1
− n(n 1)
ZMn(cid:13) − (cid:13)
then Mn is homeomorphic to t(cid:13)he sphere Sn? (cid:13)
(cid:13) (cid:13)
2010 Mathematics Subject Classification. Primary 53C40, 53C20;Secondary 53C42.
Key Words and Phrases. Curvature tensor, Ln/2-norm of curvature, Betti numbers, δ-pinched im-
mersions, flat billinear forms, Weyl tensor.
1
topological obstructions for submanifolds in low codimension
In the present paper, we provide integral curvature bounds in terms of the Betti
numbers, for compact submanifolds of Euclidean space with low codimension. As a con-
sequence, we obtain partial answers to the above problem and extend previous ones given
in [30]. Throughout the paper, all manifolds under consideration are assumed to be
without boundary, connected and oriented. Our main result is stated as follows.
Theorem 1. Given an integer n 4 and δ (1/n,1), there exists a positive constant
≥ ∈
c(n,δ) such that if Mn is a compact n-dimensional Riemannian manifold that admits an
isometric immersion f in Rn+k,2 k n/2, then
≤ ≤
n−k
scal n/2
R R dM + S δn2H2 n/2 dM c(n,δ) β (Mn;F),
− n(n 1) 1 − + ≥ i
ZMn(cid:13) − (cid:13) ZMn(cid:0) (cid:1) Xi=k
(cid:13) (cid:13)
(cid:13) (cid:13)
where S is the squared norm of second fundamental form, H the mean curvature1 of f,
S δn2H2 = max S δn2H2,0 and β (Mn;F) the i-th Betti number of Mn over an
− + { − } i
arbitrary coefficient field F. Furthermore,
(cid:0) (cid:1)
(i) If
scal n/2
R R dM + S δn2H2 n/2 dM < c(n,δ), (1)
− n(n 1) 1 − +
ZMn(cid:13) − (cid:13) ZMn
then Mn has(cid:13)the homotopy ty(cid:13)pe of a CW-com(cid:0)plex with n(cid:1)o cells of dimension i for
(cid:13) (cid:13)
k i n k. Moreover, if k = 2, then the fundamental group π (Mn) is a free
1
≤ ≤ −
group on β (Mn;Z) generators and if π (Mn) is finite then Mn is homeomorphic to
1 1
Sn.
(ii) If the scalar curvature of Mn is everywhere non-positive, then
n
scal n/2
R R dM + S δn2H2 n/2 dM c(n,δ) β (Mn;F).
1 i
− n(n 1) − ≥
ZMn(cid:13) − (cid:13) ZMn(cid:0) (cid:1) Xi=0
(cid:13) (cid:13)
(cid:13) (cid:13)
(iii) If the scalar curvature is everywhere non-positive and
scal n/2
R R dM + S δn2H2 n/2 dM < 3c(n,δ)
1
− n(n 1) −
ZMn (cid:13) − (cid:13) ZMn
(cid:0) (cid:1)
(cid:13) (cid:13)
then Mn is (cid:13)homeomorphic to(cid:13)Sn.
In the case where (1) is satisfied, the homology groups of Mn must satisfy the con-
dition H (Mn;F) = 0 for all k i n k, where F is any coefficient field.
i
≤ ≤ −
The idea of the proof is to relate the Ln/2-norm of the tensor R scal/n(n 1) R
1
− −
with the Betti numbers using Morse theory, Chern-Lashof results [11,12] and the Gauss
(cid:0) (cid:1)
1The mean curvature is given by H = H, where H denotes the mean curvature vector.
| |
2
christos-raent onti and theodoros vlachos
equation. To this aim we prove an algebraic inequality for symmetric bilinear forms (see
Prop. 9). The presence of the integral S δn2H2 n/2 dM in Theorem 1 is essential
Mn − +
since the algebraic inequality fails by dropping the corresponding term.
R (cid:0) (cid:1)
The above integral measures how far an immersion deviates from being δ-pinched.
The latter means that the inequality S δn2H2 holds everywhere, in which case δ
≤ ≥
1/n. We note that Shiohama and Xu [26,27] gave a topological lower bound of the
above integral in the case where δ = 1/n. The geometry and the topology of δ-pinched
immersions have been studied by several authors (see [1,2,2,3,6–8]) in the case where δ =
1/(n 1). Our results provide information on δ-pinched immersions for any δ (1/n,1).
− ∈
Indeed, the following corollary follows immediately from Theorem 1 and gives an intrinsic
obstruction to δ-pinched immersions.
Corollary 2. If a compact n-dimensional Riemannian manifold Mn,n 4, admits an
≥
isometric δ-pinched immersion in Rn+k,2 k n/2, for some δ (1/n,1), then
≤ ≤ ∈
n−k
scal n/2
R R dM c(n,δ) β (Mn;F).
1 i
− n(n 1) ≥
ZMn(cid:13) − (cid:13) Xi=k
(cid:13) (cid:13)
(cid:13) (cid:13)
In particular, if
scal n/2
R R dM < c(n,δ),
1
− n(n 1)
ZMn (cid:13) − (cid:13)
then Mn has the homotopy t(cid:13)ype of a CW-com(cid:13)plex with no cells of dimension i for k
(cid:13) (cid:13) ≤
i n k. Moreover, if k = 2, then π (Mn) is a free group on β (Mn;Z) generators and
1 1
≤ −
if π (Mn) is finite then Mn is homeomorphic to Sn.
1
The next results are easy consequences of Theorem 1 and provide partial answers to
the problem raised by Shiohama and Xu.
Corollary 3. If a compact n-dimensional Riemannian manifold Mn, n 4, admits an
≥
isometric immersion f in Rn+k,2 k n/2, such that
≤ ≤
scal n/2
R R dM < λ c(n,δ)
1
− n(n 1)
ZMn(cid:13) − (cid:13)
(cid:13) (cid:13)
and (cid:13) (cid:13)
n−k
S δn2H2 n/2 dM (1 λ) c(n,δ) β (Mn;F),
− + ≤ − i
ZMn i=k
(cid:0) (cid:1) X
where λ (0,1) and δ (1/n,1), then f is δ-pinched and Mn has the homotopy type of
∈ ∈
a CW-complex with no cells of dimension i for k i n k. Furthermore,
≤ ≤ −
(i) If k = 2, then π (Mn) is a free group on β (Mn;Z) generators and if π (Mn) is
1 1 1
finite then Mn is homeomorphic to Sn.
3
topological obstructions for submanifolds in low codimension
(ii) If the mean curvature is everywhere positive and δ = 1/(n 1), then Mn is diffeo-
−
morphic to Sn.
Corollary 4. If a compact n-dimensional Riemannian manifold Mn, n 4, admits an
≥
isometric immersion in Rn+k,2 k n/2, such that
≤ ≤
n−k
scal n/2
R R dM λ c(n,δ) β (Mn;F)
1 i
− n(n 1) ≤
ZMn(cid:13) − (cid:13) Xi=k
(cid:13) (cid:13)
(cid:13) (cid:13)
and
S δn2H2 n/2 dM < (1 λ) c(n,δ),
− + −
ZMn
(cid:0) (cid:1)
where λ (0,1) and δ (1/n,1), then Mn is isometric to a constant curvature sphere.
∈ ∈
Minimal submanifolds with pinched second fundamental form have been studied by
Simons [28], Chern, do Carmo, Kobayashi [9] and Leung [16], among others. We provide
intrinsic obstructions forminimal submanifolds inspheres withsufficiently pinched second
fundamental form.
Corollary 5. Let f: Mn Sn+k−1,2 k n/2, be an isometric minimal immersion
→ ≤ ≤
of a compact n-dimensional Riemannian manifold Mn,n 4. If the squared norm of the
≥
second fundamental form satisfies S n(δn 1) for some δ (1/n,1), then
≤ − ∈
n−k
scal n/2
R R dM c(n,δ) β (Mn;F).
1 i
− n(n 1) ≥
ZMn(cid:13) − (cid:13) Xi=k
(cid:13) (cid:13)
(cid:13) (cid:13)
In particular, if
scal n/2
R R dM < c(n,δ),
1
− n(n 1)
ZMn(cid:13) − (cid:13)
then Mn has the homotopy t(cid:13)ype of a CW-com(cid:13)plex with no cells of dimension i for k
(cid:13) (cid:13) ≤
i n k. Moreover, if k = 2, then the fundamental group π (Mn) is a free group on
1
≤ −
β (Mn;Z) generators and if π (Mn) is finite then Mn is homeomorphic to Sn.
1 1
It is well known that the Weyl tensor of a n-dimensional Riemannian manifold
W
Mn,n 4, vanishes if and only if Mn is conformally flat. The study of conformally
≥
flat manifolds, from the point of view of submanifold theory, was initiated by Cartan
in [4]. The case of compact conformally flat hypersurfaces of Euclidean space has been
studied by Do Carmo, Dajczer and Mercuri [13]. For low codimension k, Moore [19]
proved that such submanifolds have the homotopy type of a CW-complex with no cells
of dimension i, k < i < n k. Therefore, it is natural to seek for restrictions on the
−
topology of compact almost conformally flat submanifolds, in the sense that the Weyl
tensor is sufficiently small in a suitable norm.
4
christos-raent onti and theodoros vlachos
Thecaseofhypersurfaceshasbeentreatedin[2]. Inthispaper, weproveaninequality
for the Ln/2-norm of the Weyl tensor for compact n-dimensional Riemannian manifolds
that allow conformal immersions in the Euclidean space with low codimension. As a
consequence, we obtain a partial answer to the above question.
Theorem 6. Given n 6 and δ (1/n,1), there exists a positive constant c (n,δ) such
1
≥ ∈
that if Mn is a compact n-dimensional Riemannian manifold that admits a conformal
immersion in Rn+k,2 k [(n 2)/2], then
≤ ≤ −
n−k−1
n/2 dM + S δn2H2 n/2 dM c (n,δ) β (Mn;F).
kWk − + ≥ 1 i
ZMn ZMn i=k+1
(cid:0) (cid:1) X
In particular, if
n/2 dM + S δn2H2 n/2 dM < c (n,δ),
kWk − + 1
ZMn ZMn
(cid:0) (cid:1)
then Mn has the homotopy type of a CW-complex with no cells of dimension i for k <
i < n k.
−
As an application of Theorem 6, we may obtain results similar to Corollaries 2-5 for
the Weyl tensor instead of the tensor R scal/n(n 1) R . For instance, we have the
1
− −
following
(cid:0) (cid:1)
Corollary 7. If a compact n-dimensional Riemannian manifold Mn,n 6, admits a
≥
conformal δ-pinched immersion in Rn+k,2 k [(n 2)/2], for some δ (1/n,1), then
≤ ≤ − ∈
n−k−1
n/2 dM c (n,δ) β (Mn;F).
1 i
kWk ≥
ZMn i=k+1
X
In particular, if
n/2 dM < c (n,δ),
1
kWk
ZMn
then Mn has the homotopy type of a CW-complex with no cells of dimension i for k <
i < n k.
−
2 Algebraic auxiliary results
This section is devoted to some algebraic results that are crucial for the proofs. Let
V and W be finite dimensional real vector spaces equipped with non-degenerate inner
products which, by abuse of notation, are both denoted by , . The inner product of V
h· ·i
is assumed to be positive definite. We denote by Hom(V V,W) the space of all bilinear
×
5
topological obstructions for submanifolds in low codimension
forms and by Sym(V V,W) its subspace that consists of all symmetric bilinear forms.
×
The space Sym(V V,W) can be viewed as a complete metric space with respect to the
×
usual Euclidean norm .
k·k
The Kulkarni-Nomizu product of two bilinear forms φ,ψ Hom(V V,R) is the
∈ ×
(0,4)-tensor φ7ψ: V V V V R defined by
× × × →
φ7ψ(x ,x ,x ,x ) = φ(x ,x )ψ(x ,x )+φ(x ,x )ψ(x ,x )
1 2 3 4 1 3 2 4 2 4 1 3
φ(x ,x )ψ(x ,x ) φ(x ,x )ψ(x ,x ).
1 4 2 3 2 3 1 4
− −
Using the inner product of W, we extend the Kulkarni-Nomizu product to bilinear
forms β,γ Hom(V V,W), as the (0,4)-tensor β7γ: V V V V R defined by
∈ × × × × →
β 7γ(x ,x ,x ,x ) = β(x ,x ),γ(x ,x ) β(x ,x ),γ(x ,x )
1 2 3 4 1 3 2 4 1 4 2 3
h i−h i
+ β(x ,x ),γ(x ,x ) β(x ,x ),γ(x ,x ) .
2 4 1 3 2 3 1 4
h i−h i
A bilinear form β Hom(V V,W) is called flat with respect to the inner product
∈ ×
of W if
β(x ,x ),β(x ,x ) β(x ,x ),β(x ,x ) = 0
1 3 2 4 1 4 2 3
h i−h i
for all x ,x ,x ,x V, or equivalently if β 7β = 0.
1 2 3 4
∈
Associated to each bilinear form β is the nullity space (β) defined by
N
(β) = x V : β(x,y) = 0 for all y V .
N { ∈ ∈ }
We need the following lemma, which was given in [30, Lemma 2.1].
Lemma 8. Let β Sym(V V,W) be a bilinear form, where V and W are both equipped
∈ ×
with positive definite inner products and dimW dimV 2. If β 7β = µ , 7 ,
≤ − h· ·i h· ·i
for some µ = 0, then µ > 0 and there exist a unit vector ξ W and a subspace V V
1
6 ∈ ⊆
such that
dimV dimV dimW +1
1
≥ −
and
β(x,y) = √µ x,y ξ, for all x V and y V.
1
h i ∈ ∈
We define the map scal: Sym(V V,W) R by
× →
scal(β) = trace Ric(β),
where
1
Ric(β)(x,y) = trace R(β)( ,x, ,y), x,y V and R(β) = β 7β.
· · ∈ 2
Hereafter, we assume that V and W are both endowed with positive definite inner
products. For each β Sym(V V,W), we define the map
∈ ×
β♯: W End(V), ξ β♯(ξ)
→ 7→
6
christos-raent onti and theodoros vlachos
such that
β♯(x),y = β(x,y),ξ , for all x,y V,
h i h i ∈
where End(V) denotes the set of all selfadjoint endomorphisms of V.
Let dimV = n and dimW = k. When 2 k n/2, for each β Sym(V V,W),
≤ ≤ ∈ ×
we denote by Φ(β) the subset of the unit (k 1)-sphere Sk−1 in W given by
−
Φ(β) = u Sk−1 : k Index β♯(u) n k .
{ ∈ ≤ ≤ − }
The following proposition is crucial for the proof of Theorem 1.
Proposition 9. Given integers 2 k n/2 and λ (1/n,1), there exists a positive
≤ ≤ ∈
constant ε(n,k,λ) > 0, such that the following inequality holds
1 scal(β) 2 4/n
β7β , 7 , + β 2 λ traceβ 2 2 ε(n,k,λ) detβ♯(u) dS
4 −n(n 1)h· ·i h· ·i k k − | | + ≥ | | u
(cid:13) − (cid:13) (cid:16)ZΛ(β) (cid:17)
(cid:0) (cid:1) (2)
(cid:13) (cid:13)
(cid:13) (cid:13)
for any β Sym(V V,W), where
∈ ×
Φ(β), if scal(β) > 0
Λ(β) =
(Sk−1, if scal(β) 0.
≤
Proof: We consider the functions φ ,ψ: Sym(V V,W) R defined by
λ
× →
1 scal(β) 2
φ (β) = β 7β , 7 , + β 2 λ trace β 2 2
λ 4 − n(n 1)h· ·i h· ·i k k − | | +
(cid:13) − (cid:13)
(cid:0) (cid:1)
(cid:13) (cid:13)
and (cid:13) (cid:13)
ψ(β) = detβ♯(u) dS .
u
| |
ZΛ(β)
We shall prove that φ attains a positive minimum on Σ , where
λ n,k
Σ = β Sym(V V,W) : ψ(β) = 1 .
n,k
{ ∈ × }
There exists a sequence β in Σ such that
m n,k
{ }
lim φ (β ) = infφ (Σ ) 0.
λ m λ n,k
m→∞ ≥
Weobservethatβ = 0forallm N, sinceβ Σ . Thenwemaywriteβ = β β ,
m m n,k m m m
6 ∈ ∈ k k
where β = 1.
m
k k
We claim that the sequence β is bounded. Assume to the contrary that thbere
m
{ }
exists a bsubsequence of β , which by abuse of notation is again denoted by β , such
m m
{ } { }
that lim β = . Since β = 1, we may assume, by taking a subsequence
m→∞ m m
k k ∞ k k
if necessary, that β converges to some β Sym(V V,W) with β = 1. Using
m
{ } ∈ × k k
b
b 7b b
topological obstructions for submanifolds in low codimension
the fact that φ is homogeneous of degree 4, we have φ (β ) = φ (β )/ β 4. Thus
λ λ m λ m m
k k
lim φ (β ) = 0 and consequently φ (β) = 0, or equivalently
m→∞ λ m λ
b
b scabl(β)
β 7β = , 7 ,
n(n 1)h· ·i h· ·i
−
b
b b
and
1 λ trace β 2 = λ(scal(β)+1). (3)
≤ | |
Since λ < 1, equation (3) implies scal(β) > 0. According to Lemma 8, there exists a unit
b b
vector ξ W and subspace V of V with dimV n k +1 such that
1 1
∈ ≥ −
b
b scbal(β) 1/2 b
β(x,y) = x,y ξ for all x V and y V. (4)
1
n(n 1) h i ∈ ∈
(cid:16) −b (cid:17)
b b b
Moreover, since β is in Σ , there exists an open subset of Sk−1 such that
m n,k m m
{ } U U ⊆
Λ(β ) and detβ♯ (u) = 0 for all u and m N. From scal(β) > 0, we deduce that
m m 6 ∈ Um ∈
scal(β ) > 0 and so Φ(β ) for m large enough. b b
m m m
U ⊆
bLet u bbe a sequence such thabt u for all m N.b We may assume that
m m m
{ } ∈ U ∈
u bis convergent, bby passinbg if necessary to a subsequence and set u = lim u .
m m→∞ m
{ }
Since limmb→∞βm♯ (um) = β♯(u) and um ∈bUm, webdeduce that Index β♯(u) ≤ n−k. Then,
frbom (4) we obtain ξ,u 0. We claim that ξ,u = 0. Indeed, if ξ,bu > 0, thenb(4)
h i ≥ h i h i
implies that β♯b(u)bhas atbleabst n kb+1 pbositive eigenvalues and so,bfobr m large enough,
−
β♯ (u ) has at least nb bk +1 positive eigenvalubesb. This and the fact tbhabt detβ♯ (u) = 0
m m − m 6
for all u b,bshows that β♯ (u ) has at most k 1 negative eigenvalues. Therefore,
∈ Um m m −
Ibndebx β♯ (u ) k 1 which is a contradiction, since u . b
m m ≤ − m ∈ Um
Thus, wbe have proved thbat fbor any convergent sequence u such that u for
m m m
{ } ∈ U
all m, wbe hbave lim u ,ξ = 0. b b
m→∞ m
h i
Since is open, we may choose convergent sequences bu(1) , u(2) , b , u(bk) in
m m m m
U { } { } ··· { }
such that u(1),u(2), b ,bu(k) span W for all m N. From (4) and the fact that
m m m m
U ··· ∈
lim u(ba),ξ = 0 for all a 1,2, ,k , we obtain thatbthe rebstriction ofbβ to
m→∞ m m
h i ∈ { ··· }
Vb V satisfiesb b b
1 1
×
b b mli→m∞βm|Vb1×Vb1 = 0 b
b b
and consequently
b
mli→m∞(βm 7βm)|Vb1×Vb1×Vb1×Vb1 = 0. (5)
From the inequality
b b
1 scal(β ) 2
β 7β m , 7 , φ (β ),
4 m m − n(n 1)h· ·i h· ·i Vb1×Vb1×Vb1×Vb1 ≤ λ m
(cid:13)(cid:16) −b (cid:17)(cid:12) (cid:13)
(cid:13) (cid:13)
b b (cid:12) b
(cid:13) 8 (cid:13)
christos-raent onti and theodoros vlachos
(5) and the fact that lim φ (β ) = 0 we obtain scal(β) = 0, which contradicts (3).
m→∞ λ m
Thus, the sequence β is bounded, and it converges to some β Sym(V V,W), by
m
{ } ∈ ×
taking a subsequence if necessary.b b
We claim that φ (β) > 0. Arguing indirectly, we assume that φ (β) = 0. Then
λ λ
scal(β)
β 7β = , 7 ,
n(n 1)h· ·i h· ·i
−
and
β 2 λ trace β 2 = λ(scal(β)+ β 2). (6)
k k ≤ | | k k
We notice that β = 0. Indeed, if β = 0, then β♯(u) = 0 for all u Sk−1. Since β Σ
m n,k
6 ∈ ∈
for all m N, there exists ξ Λ(β ) such that
m m
∈ ∈
detβ♯ (ξ ) Vol(Λ(β )) = 1 for all m N. (7)
| m m | m ∈
We may assume that ξ converges to some ξ, by passing to a subsequence if necessary.
m
Then lim β♯ (ξ ) = β♯(ξ) = 0, which contradicts (7). Therefore β = 0.
m→∞ m m 6
Now, from (6) we obtain that scal(β) = 0. Then, Lemma 8 implies that scal(β) > 0
6
and there exists a unit vector ξ W and a subspace V of V with dimV n k + 1
1 1
∈ ≥ −
such that
scal(β) 1/2
β(x,y) = x,y ξ, for all x V and y V. (8)
1
n(n 1) h i ∈ ∈
(cid:16) − (cid:17)
Since β Σ for all m N, there exists an open subset of Sk−1 such that
m n,k m
∈ ∈ U
Λ(β ) and detβ♯ (u) = 0, for all u and m N. Moreover, we have that
Um ⊆ m m 6 ∈ Um ∈
scal(β ) > 0 and so Φ(β ) for m large enough.
m m m
U ⊆
Let u be a sequence such that u for all m N. We may assume that
m m m
{ } ∈ U ∈
u is convergent, by passing if necessary to a subsequence and set u = lim u . Since
m m→∞ m
lim β♯ (u ) = β♯(u) and u it follows that Index β♯(u) n k. Then, from
m→∞ m m m ∈ Um ≤ −
(8) we get ξ,u 0. We claim that ξ,u = 0. Indeed, if ξ,u > 0 then (8) implies
h i ≥ h i h i
that β♯(u) has at least n k+1 positive eigenvalues and so, for m large enough, β♯ (u )
− m m
has at least n k +1 positive eigenvalues. This and the fact that detβ♯ (u) = 0 for all
− m 6
u and m N, shows that β♯ (u ) has at most k 1 negative eigenvalues, which is
∈ Um ∈ m m −
a contradiction, since u for all m N.
m m
∈ U ∈
Thus, we have proved that for any convergent sequence u such that u for
m m m
{ } ∈ U
allm, we have lim u ,ξ = 0.Since is open, we maychoose convergent sequences
m→∞ m m
h i U
u(1) , u(2) , , u(k) in such that u(1),u(2), ,u(k) span W for all m N. Then,
m m m m m m m
{ } { } ··· { } U ··· ∈
(a)
from (8) and the fact that lim u ,ξ = 0 for all a 1,2, ,k we obtain that
m→∞ m
h i ∈ { ··· }
the restriction of β to V V satisfies
m 1 1
×
lim β = 0
m→∞ m|V1×V1
and consequently
lim (β 7β ) = 0. (9)
m→∞ m m |V1×V1×V1×V1
9
topological obstructions for submanifolds in low codimension
From the inequality
1 scal(β ) 2
β 7β m , 7 , φ (β ),
4 m m − n(n 1)h· ·i h· ·i V1×V1×V1×V1 ≤ λ m
(cid:13)(cid:16) − (cid:17) (cid:13)
(cid:12)
(9) and the fac(cid:13)(cid:13)t that limm→∞φλ(βm) = 0 we obta(cid:12)in scal(β) =(cid:13)(cid:13)0, which contradicts (6).
Thus, we have proved that φ (β) > 0 and so φ attains a positive minimum on Σ which
λ λ n,k
obviously depends only on n,k and λ and is denoted by ε(n,k,λ).
Now, let β Sym(V V,W). Assume that ψ(β) = 0 and set γ = β/(ψ(β))1/n.
∈ × 6
Clearly γ Σ , and consequently φ (γ) ε(n,k,λ). Since φ is homogeneous of degree
n,k λ λ
∈ ≥
4, the desired inequality is obviously fulfilled. In the case where ψ(β) = 0, the inequality
is trivial.
We also need the following result on flat bilinear forms, which is due to Moore [19,
Proposition 2].
Lemma 10. Let β Sym(V V,U) be a flat bilinear form with respect to a Lorentzian
∈ ×
inner product of U. If dimV > dimU and β(x,x) = 0 for all non-zero x V, then there
6 ∈
is a non-zero isotropic vector e U and a bilinear form φ Sym(V V,R) such that
∈ ∈ ×
dim (β eφ) dimV dimU +2.
N − ≥ −
We define the map W: Sym(V V,W) Hom(V V V V,R) by
× → × × ×
W(β) = R(β) L(β)7 , ,
− h· ·i
where
1 scal(β)
L(β) = Ric(β) , ,
n 2 − 2(n 1)h· ·i
− (cid:16) − (cid:17)
The following lemma is in fact contained in [19]. For the sake of completeness we
give a short proof.
Lemma 11. Let β Sym(V V,W) be a bilinear form and dimW < dimV 2. If
∈ × −
W(β) = 0, then there exists a vector ξ W and a subspace V V such that
1
∈ ⊆
dimV dimV dimW
1
≥ −
and
β(x,y) = x,y ξ, for all x V and y V.
1
h i ∈ ∈
Proof: We endow the vector space W = W R2 with the Lorentzian inner product
⊕
, given by
hh· ·ii
(ξ,(s ,s )),(η,(ft ,t )) = ξ,η +s t +s t
1 2 1 2 1 2 2 1
hh ii h i
and define the symmetric bilinear form β: V V W by
× →
β(x,y) = β(x,y), x,y , L(β)(x,y) .
e f
h i −
(cid:0) (cid:1)
e 10