Geometric and differentiable rigidity of submanifolds in spheres

J. Math. Pures Appl. 99 (2013) 330–342

www.elsevier.com/locate/matpur

Geometric and differentiable rigidity of submanifolds in spheres

Hongwei Xu, Fei Huang, Entao Zhao

Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China Received 31 March 2012

Available online 12 October 2012

Abstract

In this paper, we investigate rigidity of geometric and differentiable structures of complete submanifolds via an extrinsic geo- metrical quantityτ (x)defined by the second fundamental form. We verify a geometric rigidity theorem for complete submanifolds with parallel mean curvature in a unit sphereS^n+p. Inspired by the rigidity theorem, we prove a differentiable sphere theorem for complete submanifolds inS^n+p. Moreover, we obtain a differentiable pinching theorem for complete submanifolds in aδ(>¹₄)- pinched Riemannian manifold.

©2012 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article, on étudie la rigidité de structures géométriques et différentiable de sous-variétés complètes via une quantité géométrique extrinsèqueτ (x)définie par sa deuxième forme fondamentale. On démonte un théorème de rigidité géométrique pour les sous-variétés complètes à courbure moyenne parallèle dans une sphère unitéS^n+p. Inspiré par le théorème de rigidité, on établit un théorème de sphère différentiable pour les sous-variétés complètes dansS^n+p. On obtient aussi un théorème de pincement différentiable pour les sous-variétés complètes dans une variété riemannienneδ(>¹₄)-pincée.

©2012 Elsevier Masson SAS. All rights reserved.

MSC:53C20; 53C24; 53C40

Keywords:Complete submanifolds; Geometric and differentiable structures; Rigidity theorem; Ricci flow; Stable currents

1. Introduction

The study of relations between geometric invariants and the structures of the geometry and topology of a manifold is a longstanding subject in global differential geometry (see[2,5,8,24]). In 1930’s, Hopf conjectured that a compact, simply connected Riemannian manifold whose sectional curvatures are close to 1 is homeomorphic to a sphere. This conjecture is known as the curvature pinching problem, which was first taken up by Rauch[23]. In 1960’s, making use of the comparison technique, Berger[1]and Klingenberg[16]proved that a compact, simply connected manifold M whose sectional curvatures all lie in the interval (1/4,1]is homeomorphic to the sphereSⁿ. Since the complex projective spaceCP^m(m2)has sectional curvatures in the interval[1/4,1], the pinching constant 1/4 is optimal

* Corresponding author.

E-mail addresses:[email protected](H. Xu),[email protected](F. Huang),[email protected](E. Zhao).

0021-7824/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.matpur.2012.09.003

for even dimensional cases. Later, Micallef and Moore [20]obtained a topological sphere theorem for pointwise 1/4-pinched manifolds via the technique of minimal surfaces. Meanwhile, the geometry of compact Kähler manifolds admitting negatively 1/4-pinched Riemannian metric was investigated by Yau and Zheng[34]. In 2008, Böhm and Wilking [3] proved that a compact Riemannian manifold with 2-positive curvature operator is diffeomorphic to a spherical space form. Recently, Brendle and Schoen[6] proved a remarkable differentiable pinching theorem for pointwise 1/4-pinched Riemannian manifolds by developing Hamilton’s Ricci flow theory[15]. Furthermore, they gave a classification of weakly pointwise 1/4-pinched Riemannian manifolds[7]. More recently, Brendle[4]obtained the celebrated convergence result for the Ricci flow.

Theorem 1.1.Let(M, g₀)be a compact Riemannian manifold of dimensionn4. Assume that

R₁₃₁₃+λ²R₁₄₁₄+R₂₃₂₃+λ²R₂₄₂₄−2λR1234>0, (1) for all orthonormal four frames{e₁, e₂, e₃, e₄}and allλ∈ [−1,1]. Then the normalized Ricci flow with initial met- ricg₀

∂

∂tg(t)= −2Ricg(t)+2

nrg(t)g(t),

exists for all time and converges to a metric with positive constant sectional curvature ast→ +∞. Herer_g(t)denotes the mean value of the scalar curvature ofg(t).

The purpose of the present article is to investigate the geometry and topology from the viewpoint of submanifolds.

After the pioneering rigidity theorem for closed minimal submanifolds in a unit sphere due to Simons[26], Lawson [17]and Chern, do Carmo and Kobayashi[12]obtained a classification ofn-dimensional closed minimal submanifolds inS^n+p whose squared norm of the second fundamental form satisfiesSn/(2−1/p). Further discussions on the rigidity theorems have been carried out by many other authors (see[21,22,33], etc.). In 1990, Xu[29]proved a rigidity theorem for compact submanifolds with parallel mean curvature in a sphere.

Theorem 1.2.LetMbe ann-dimensional oriented compact submanifold with parallel mean curvature in an(n+p)- dimensional unit sphereS^n+p. Denote byHandSthe mean curvature and squared norm of the second fundamental form, respectively. IfSC(n, p, H ),thenMis either a totally umbilic sphereSⁿ(√ ¹

1+H²), a Clifford hypersurface in an(n+1)-sphere, or the Veronese surface inS⁴(√¹

1+H²). Here the constantC(n, p, H )is defined by

C(n, p, H )=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

α(n, H ), forp=1, orp=2andH=0,

n 2−p¹

, forp2andH=0,

min{α(n, H ),^n+nH2

2−p−¹1

+nH²}, forp3andH=0, α(n, H )=n+ n³

2(n−1)H²−n(n−2) 2(n−1)

n²H⁴+4(n−1)H².

Later, the above pinching constantC(n, p, H )was improved, by Li and Li[19]forH =0 and by Xu[30]for H=0, to

C(n, p, H )=

α(n, H ), forp=1, orp=2 andH=0, min{α(n, H ),¹₃(2n+5nH²)}, otherwise.

The special case of Theorem1.2forp=1 was also studied by Cheng and Nakagawa[11]independently.

On the other hand, Lawson and Simons[18]proved a topological sphere theorem for closed submanifolds in a unit sphere by using the non-existence for stable currents on compact submanifolds in a sphere. Shiohama and Xu[25]

improved Lawson–Simons’ result and proved a topological sphere theorem for complete submanifolds in a simply connected space form with nonnegative constant curvature. Recently, Xu and Zhao[32]and Xu and Gu[31]obtained some differentiable sphere theorems for complete submanifolds.

LetMⁿbe ann-dimensional Riemannian submanifold in an (n+p)-dimensional Riemannian manifoldN^n+p. Set U M=

x∈MU_xM, whereU_xM= {u∈T_xM: u =1}andT_xMis the tangent space atx∈M. In 1986, Gauchman [13]proved that ifMis ann-dimensional closed minimal submanifold inS^n+p, and ifσ (u)¹₃ for any unit vector u∈U M, whereσ (u)= h(u, u) ²andhis the second fundamental form ofM, then eitherσ (u)≡0 orσ (u)≡¹₃. Xu, Fang and Xiang[28]generalized this rigidity result to the case whereMis ann-dimensional closed submanifold with parallel mean curvature inS^n+p. In[32], Xu and Zhao proved the following theorem.

Theorem 1.3. LetM be an n-dimensional complete submanifold in an (n+p)-dimensional unit sphere S^n+p. If σ (u) <¹₃ for allu∈U M, thenMis diffeomorphic to the standard unitn-sphereSⁿ.

Puttingβ(u, v)= h(u, u)−h(v, v) ²foru, v∈UxM, we define τ (x)= max

u,v∈UxM, u⊥vβ(u, v).

Note that if h(u, u) ²< cfor anyu∈U_xM, wherecis some positive constant, thenβ(u, v) <4cfor anyu, v∈ UxM. This impliesτ (x) <4c.

In this paper, we first obtain the following geometric rigidity theorem.

Theorem A. Let M be an n-dimensional complete submanifold with parallel mean curvature in an (n+p)- dimensional unit sphereS^n+p.

(1) Whenp=1, orp=2andH=0, ifMis compact andτ (x)4for allx∈M,then eitherτ (x)≡0andMis the totally umbilical sphereSⁿ(√¹

1+H²), orτ (x)≡4.

(2) Whenp3, orp=2andH=0, ifτ (x)⁴₃ for allx∈M,then eitherτ (x)≡0andMis the totally umbilical sphereSⁿ(√¹

1+H²), orτ (x)≡⁴₃.

Corollary 1.4.(See[13,28].) LetMbe ann-dimensional complete submanifold with parallel mean curvature in an (n+p)-dimensional unit sphereS^n+p.

(1) Whenp=1, orp=2andH=0, ifMis compact andσ (u)1for allu∈U M,then eitherσ (u)≡0andMis the totally umbilical sphereSⁿ(√¹

1+H²), ormaxu∈U Mσ (u)=1.

(2) Whenp3, orp=2andH=0, ifσ (u)¹₃for allu∈U M,then eitherσ (u)≡0andMis the totally umbilical sphereSⁿ(√¹

1+H²), orσ (u)≡¹₃.

Then we prove the following differentiable rigidity theorem, which is a generalization of Theorem1.3.

Theorem B. Let M be an n-dimensional complete submanifold in an (n+p)-dimensional unit sphereS^n+p. If τ (x) <⁴₃for allx∈M, thenMis diffeomorphic to the standard unitn-sphereSⁿ.

Remark 1.5.Examples3.5and3.6in Section3show that our pinching constants in Theorems A and B are sharp.

If the ambient space is a general Riemannian manifold, we obtained the following differentiable pinching theorem.

Theorem C. Let M be an n-dimensional complete submanifold in an (n+p)-dimensional pointwise δ(>1/4)- pinched Riemannian manifold N^n+p. Denote by K(x, π )the sectional curvature ofN for 2-plane π⊂TxN and pointx∈N. SetKmax(x):=maxπ⊂T_xNK(x, π )forx∈M. Ifτ (x) <¹⁶₉Kmax(x)(δ−¹₄)for allx∈M,thenMis diffeomorphic to a space form. In particular, ifMis simply connected, thenMis diffeomorphic to the standard unit n-sphereSⁿor the Euclidean spaceRⁿ.

Our paper is organized as follows. In Section2, we introduce some basic equations in the geometry of submanifolds and give a lower bound for the sectional curvature of the submanifold in terms of the extrinsic quantityτ (x). We prove two rigidity theorems for submanifolds of spheres in Section3. Then we give the proof of Theorem A. In Section4, Theorems B and C are proved by using convergence results for the Ricci flow of Hamilton[15]and Brendle[4], and the non-existence theorem for stable currents due to Lawson and Simons[18]. Moreover, we present a differentiable pinching theorem for even dimensional submanifolds in pinched Riemannian manifolds.

2. Preliminaries

LetMbe ann-dimensional Riemannian manifold isometrically immersed in an (n+p)-dimensional Riemannian manifoldN^n+p. We shall make use of the following convention on the range of indices:

1A, B, C, . . .n+p, 1i, j, k, . . .n, n+1α, β, γ , . . .n+p.

Choose a local orthonormal frame field{e_A}onN^n+p such thate_i’s are tangent toM. Let{ω_A}be the dual frame field of{e_A}and{ω_AB}the connection 1-forms ofN^n+p. Restricting these forms toM, we have

ω_αi=

j

h^α_ijω_j, h^α_ij =h^α_{j i},

h=

α,i,j

h^α_ijωi⊗ωj⊗eα, ξ=1

n α,i

h^α_iie_α,

R_{ij kl}=K_{ij kl}+

α

h^α_ikh^α_{j l}−h^α_ilh^α_{j k}

, (2)

R_αβkl=K_αβkl+

i

h^α_ikh^β_il−h^α_ilh^β_ik

, (3)

whereh,ξ,R_{ij kl},R_αβklare the second fundamental form, the mean curvature vector, the Riemannian curvature tensor and the normal curvature tensor ofM, andK_ABCD is the Riemannian curvature tensor ofN, respectively. We set

H= ξ , H^α= h^α_ij

n×n.

Denote the first and second covariant derivatives ofh^α_ij byh^α_{ij k}andh^α_{ij kl}, respectively. Then we have

k

h^α_{ij k}ω_k=dh^α_ij+

k

h^α_kjω_ik+

k

h^α_ikω_{j k}+

β

h^β_ijω_αβ,

k

h^α_{ij kl}ω_l=dh^α_{ij k}+

l

h^α_{lj k}ω_il+

l

h^α_ilkω_{j l}+

l

h^α_{ij l}ω_kl+

β

h^β_{ij k}ω_αβ.

The Laplacian ofhis defined byh^α_ij=

kh^α_{ij kk}. If the mean curvature vector ofMis parallel in the normal bundle T^⊥M, i.e.,∇_X^⊥ξ=0 for anyX∈Γ (T M), following[28,30,33], etc. we have

i

h^α_iik=0,

i

h^α_iikl=0, for allk, l, α, (4)

h^α_ij=

k,m

h^α_kmR_{mij k}+

k,m

h^α_miR_{mkj k}+

k,β

h^β_kiR_{βαj k}, (5)

α

Rαβkl

trH^α

=0. (6)

LetU M denote the unit tangent bundle onM andU_xM its fiber overx∈M. ThenU M=

x∈MU_xM, where U_xM= {u∈T_xM: u =1},T_xMis the tangent space ofMatx. Set

β(u, v)=h(u, u)−h(v, v)², foru, v∈UxM. We define a new extrinsic geometrical invariant ofMby

τ (x)= max

u,v∈UxM,u⊥vβ(u, v), x∈M.

Proposition 2.1.LetMbe ann-dimensional submanifold in a Riemannian manifoldN^n+p, andxa fixed point inM.

Thenτ (x)=0if and only ifxis a totally umbilical point.

Proof. A pointx∈Mis totally umbilical if and only if for eachα=n+1, . . . , n+p, the eigenvalues of the matrix H^α are equal. Suppose{e₁, . . . , e_n}is an orthonormal basis ofT_xMsuch thath(e_i, e_j), e_α =κ_i^αδ_ij. Thenκ_i^α,i= 1, . . . , n, aren eigenvalues ofH^α. Ifτ (x)=0, then h(e_i, e_i)=h(e_j, e_j). This implies thatκ_i^α = h(e_i, e_i), e_α = h(e_j, e_j), e_α =κ_j^α fori=j. That is to say,H^α has only one eigenvalue with multiplicationn. Note that this holds for arbitraryα=n+1, . . . , n+p. HenceMis totally umbilical atx. The inverse is obvious since thatMis totally umbilical atx is equivalent toh(X, Y )= X, Yξ for anyX, Y∈T_xM. 2

The following lemma gives a lower bound for the sectional curvature of a submanifold.

Lemma 2.2. Let M be an n-dimensional submanifold in a Riemannian manifold N^n+p. Denote by K(x, π ) the sectional curvature ofN for 2-plane π⊂T_xN and point x∈N. SetK_min(x):=minπ⊂TxNK(x, π ) for x ∈M.

Then the sectional curvature ofMatx∈Mis no less thanK_min(x)−^{τ (x)}₂ .

Proof. Forx∈M, letu, vbe two perpendicular unit vectors inUxM. Then by Gauss equation (2) we have R(u, v, u, v)=K(u, v, u, v)+

h(u, u), h(v, v)

−h(u, v)². (7) Note that

τ (x)h(u, u)−h(v, v)²=h(u, u)²+h(v, v)²−2

h(u, u), h(v, v) . Therefore,

h(u, u), h(v, v)

1

4h(u, u)²+h(v, v)²−τ (x) +1

2

h(u, u), h(v, v) −τ (x)

4 . (8)

On the other hand, we have

h(u, v)=1 2

h u+v

√2 ,u+v

√2

−h u−v

√2 ,u−v

√2

.

Since(u+v)/√

2 and(u−v)/√

2 are two perpendicular unit vectors, we have h(u, v)²=1

4 h

u+v

√2 ,u+v

√2

−h u−v

√2 ,u−v

√2

²τ (x)

4 . (9)

Combining (7), (8) and (9), we complete the proof of the lemma. 2 3. Rigidity of submanifolds with parallel mean curvature

In this section, we investigate the geometric rigidity of submanifolds with parallel mean curvature. Let Mⁿ be ann-dimensional complete submanifold with parallel mean curvature in an(n+p)-dimensional unit sphereS^n+p. ThenK_ABCD=δ_ACδ_BD−δ_ADδ_BC.

Supposeτ (x₀)=0 atx₀∈M. Then there exist two perpendicular unit vectorsu₀, v₀∈U_x0Msuch that

u,v∈Umaxx0M,u⊥v

h(u, u)−h(v, v)²=h(u₀, u₀)−h(v₀, v₀)².

Choose an orthonormal frame{e_A}atx₀such that

e_n+1= h(u0, u0)−h(v0, v0)

h(u₀, u₀)−h(v₀, v₀) , (10)

and the matrix(hⁿ_ij⁺¹)satisfies

hⁿ₁₁⁺¹· · ·hⁿ_nn⁺¹, hⁿ_ij⁺¹=0 (i=j ). (11) Letu0=

ixⁱei, v0=

iyⁱei. Since τ (x0)=

h(u0, u0)−h(v0, v0), en+1

2

=

i

x_i²−y²_i hⁿ_ii⁺¹

2

i

x_i²hⁿ₁₁⁺¹−

i

y²_ihⁿ_nn⁺¹ 2

=

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

,

by the definition ofτ, we haveτ (x0)=(hⁿ₁₁⁺¹−hⁿ_nn⁺¹)². Thus we can take

e₁=u₀, e_n=v₀. (12)

From (10), we know thath(e₁, e₁)−h(e_n, e_n)is parallel toe_n+1. Hence

h^α₁₁=h^α_nn, α=n+1. (13)

On the other hand, for anyx∈M,e_i, e_j∈T_xM, i=j, let u=e_i+e_j

√2 , v=e_i−e_j

√2 . Sinceu, vare perpendicular, we have

α

h^α_ij2

(x)=1

4h(u, u)−h(v, v)² (14)

1

4τ (x). (15)

Define a tensor fieldA=(Aij kl)onMby

A_{ij kl}=

α

h^α_ijh^α_kl.

Under the frame satisfying (10)–(13), we have

τ (x0)=A1111+Annnn−2A11nn. (16) By the definition of Laplacian, we obtainτ (x₀)=(1, n)=(A)₁₁₁₁+(A)_nnnn−2(A)11nn, and

1

2(A)₁₁₁₁=

α,i

h^α₁₁h^α_11ii+ h^α_11i2

,

1

2(A)nnnn=

α,i

h^α_nnh^α_nnii+ h^α_nni2

,

(A)_11nn=

α,i

h^α₁₁h^α_nnii+h^α_nnh^α_11ii+2h^α_11ih^α_nni .

Using (13), atx₀we have

1

2(1, n)=

i

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

hⁿ_11ii⁺¹−hⁿ_nnii⁺¹ +

α,i

h^α_11i−h^α_nni2

i

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

hⁿ_11ii⁺¹−hⁿ_nnii⁺¹

. (17)

Substituting (2), (3) and (5) into (17), we have 1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

−2

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

i,α

hⁿ₁₁⁺¹−hⁿ_ii⁺¹ h^α_1i2

+

hⁿ_ii⁺¹−hⁿ_nn⁺¹ h^α_ni2 +

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

i,α

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

h^α₁₁h^α_ii+

hⁿ_ii⁺¹−hⁿ_nn⁺¹ h^α_nnh^α_ii

. (18)

Lemma 3.1.LetM be ann-dimensional submanifold with parallel mean curvature in a unit sphereS^n+p. Suppose τ (x)=0atx∈M. Let{e_A}be an adapted frame satisfying(10)–(13).

(1) Ifp=1, orp=2andH=0, then 1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 1−1

4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

; (2) Ifp3, orp=2andH=0, then

1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 1−3

4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 .

Proof. (1) Ifp=1, by (10) we see that the second term of the right hand side of (18) vanishes. Ifp=2 andH=0, by (6) we have

α

R_αβkl trH^α

=0.

Letα,β ben+1 andn+2, respectively. Then we have R_(n+1)(n+2)kl

trHⁿ⁺¹

=0, R_(n+2)(n+1)kl

trHⁿ⁺²

=0.

SinceH=0, trHⁿ⁺¹and trHⁿ⁺²do not equal to 0 at the same time. Hence R_(n+1)(n+2)kl=0

for allk, l=1, . . . , n.By (3) and (11) we have R_(n+1)(n+2)kl=

n

i=1

hⁿ_ik⁺¹hⁿ_il⁺²−hⁿ_il⁺¹hⁿ_ik⁺²

=hⁿ_kk⁺¹hⁿ_kl⁺²−hⁿ_ll⁺¹hⁿ_lk⁺²

=hⁿ_kl⁺²

hⁿ_kk⁺¹−hⁿ_ll⁺¹

=0.

Particularly,

hⁿ_1i⁺²

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

=0, hⁿ_ni⁺²

hⁿ_nn⁺¹−hⁿ_ii⁺¹

=0.

So the second term of the right hand side of (18) vanishes either. Hence we have the following estimate from (18):

1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

+

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

i

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

α

h^α₁₁h^α_ii+

hⁿ_ii⁺¹−hⁿ_nn⁺¹

α

h^α_nnh^α_ii

n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

+

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

×

i

−1 4

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

−1 4

hⁿ_ii⁺¹−hⁿ_nn⁺¹

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

=n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 1−1

4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

. (19)

In the second inequality we have used the following estimate:

−

α

h^α₁₁h^α_ii

α

1 4

h^α₁₁−h^α_ii2

=1

4h(e1, e1)−h(ei, ei)²

1

4h(e₁, e₁)−h(e_n, e_n)²

=1 4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

. (20)

(2) Ifp3, orp=2 andH=0, (18) can be rewritten as 1

2(1, n)=n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

− hⁿ₁₁⁺¹

−hⁿ_nn⁺¹ _n

i=2

hⁿ₁₁⁺¹−hⁿ_ii⁺¹ 2

α

h^α_1i2

−

α

h^α₁₁h^α_ii

+

n−1

i=1

hⁿ_ii⁺¹−hⁿ_nn⁺¹ 2

α

h^α_ni2

−

α

h^α_nnh^α_ii

. (21)

Fori=1, by (15) we have

2

α

h^α_1i2

1

2

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

.

From this and (20) we obtain

n

i=2

hⁿ₁₁⁺¹−hⁿ_ii⁺¹ 2

α

h^α_1i2

−

α

h^α₁₁h^α_ii

n

i=2

hⁿ₁₁⁺¹−hⁿ_ii⁺¹

×3 4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

. (22)

Similarly,

n−1

i=1

hⁿ_ii⁺¹−hⁿ_nn⁺¹ 2

α

h^α_ni2

−

α

h^α_nnh^α_ii

n−1

i=1

hⁿ_ii⁺¹−hⁿ_nn⁺¹

×3 4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

. (23)

Substituting (22) and (23) into (21), we get

1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

−

hⁿ₁₁⁺¹−hⁿ_nn⁺¹

×3 4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

_n

i=2

hⁿ₁₁⁺¹−hⁿ_ii⁺¹ +

n−1

i=1

hⁿ_ii⁺¹−hⁿ_nn⁺¹

=n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

−3 4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹3

× _n

i=1

hⁿ₁₁⁺¹−hⁿ_ii⁺¹ +

n

i=1

hⁿ_ii⁺¹−hⁿ_nn⁺¹

=n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2

−3 4n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹4

=n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 1−3

4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 . This completes the proof of the lemma. 2

Lemma 3.2.LetMbe ann-dimensional compact submanifold with parallel mean curvature in a unit sphereS^n+p. We have

(1) Ifp=1, orp=2andH=0, andτ (x)4for anyx∈M, thenτ (x)is a constant function onM.

(2) Ifp3, orp=2andH=0, andτ (x)⁴₃for anyx∈M, thenτ (x)is a constant function onM.

Proof. It suffices to show thatτ (x)is a subharmonic function. Fix a pointx∈M. For the Laplacian of continuous functions, we have the generalized definition

(τ )(x)=Clim

r→0

1 r²

B(x,r)

τ

B(x,r)

1−τ (x)

,

whereCis a positive constant andB(x, r)is a geodesic ball centered atxwith radiusr.

Ifτ (x)=0, thenτ (x)0 sinceτ is nonnegative onM.

If τ (x)=0, in a small neighborhood O_x of x within the cut-locus ofx, we denote by u(y), v(y)two vectors tangent toMobtained by parallel transport ofu(x)=e₁, v(x)=e_nalong the unique geodesic joiningxtoy. Define f_x(y)= h(u(y), u(y))−h(v(y), v(y)) ². Then

f_x(y)|y=x=

A(u, u, u, u)+A(v, v, v, v)−2A(u, u, v, v)

y=x=(1, n).

Sinceτ (x)=fx(x)andτ (y)fx(y), we haveτ (x)fx(x). From Lemma3.1we know that(1, n)0. Hence τ (x)0.

Sincex is arbitrary, we see thatτ (x)is subharmonic, which implies thatτ is a constant function. This completes the proof of Lemma3.2. 2

Theorem 3.3.LetMbe ann-dimensional compact submanifold with parallel mean curvature in a unit sphereS^n+p. If p=1, orp=2and H=0, and ifτ (x)4for allx∈M,then either τ (x)≡0 andM is the totally umbilical sphereSⁿ(√ ¹

1+H²), orτ (x)≡4.

Proof. By Lemma3.2,τis a constant function. Whenτ =0, from Lemma2.1we know thatMis the totally umbilical sphereSⁿ(√ ¹

1+H²). Whenτ =0, from the definition off_x(y)in the proof of Lemma3.2we see thatf_x(x)is the maximal value off_x. Hence,(1, n)=f_x(x)0. By Lemma3.1we have

1

2(1, n)n

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 1−1

4

hⁿ₁₁⁺¹−hⁿ_nn⁺¹2 .

Since τ (x)=(hⁿ₁₁⁺¹−hⁿ_nn⁺¹)²4, we obtain from the above inequality thatτ (x)=4. This completes the proof of Theorem3.3. 2

Theorem 3.4.LetMbe ann-dimensional complete submanifold with parallel mean curvature in a unit sphereS^n+p. Ifp3, orp=2 andH=0, and ifτ (x)⁴₃ for allx∈M, then eitherτ (x)≡0 andMis the totally umbilical sphereSⁿ(√¹

1+H²), orτ (x)≡⁴₃.

Proof. From Lemma2.2, we haveKM¹₃>0. HenceMis a compact submanifold by Myers’ theorem. Whenτ=0, Mis the totally umbilical sphereSⁿ(√¹

1+H²). Whenτ is a nonzero constant, we have(1, n)0. From Lemma3.1 we have ¹₂(1, n)n(hⁿ₁₁⁺¹−hⁿ_nn⁺¹)²(1−³₄(hⁿ₁₁⁺¹−hⁿ_nn⁺¹)²). Hence ifτ (x)=(hⁿ₁₁⁺¹−hⁿ_nn⁺¹)²⁴₃, we obtain that τ=⁴₃. This completes the proof of Theorem3.4. 2

Now we are in a position to give the proof of Theorem A.

Proof of Theorem A. Combing Theorems3.3and3.4, we obtain the assertion of Theorem A. 2

Example 3.5. Let S^q(r) be the q-dimensional sphere of radius r in R^q+1, and let 1k n−1. We embed S^k(1/√

2)×S^n−k(1/√

2)inSⁿ⁺¹(1)as follows. Letu∈S^k(1/√

2)andv∈S^n−k(1/√

2)be vectors of length 1/√ 2 inR^k+1andR^n−k+1, respectively. We can consider(u, v)as a unit vector inRⁿ⁺²=R^k+1×R^n−k+1. It is easy to see thatS^k(1/√

2)×S^n−k(1/√

2)is a submanifold inSⁿ⁺¹(1)with constant mean curvature H=

2k−n n

. In particular,S^k(1/√

2)×S^k(1/√

2)⊂Sⁿ⁺¹(1)is minimal ifn=2k. SinceSⁿ⁺¹(1)⊂Sⁿ⁺²(1)is totally geodesic, as a submanifold inSⁿ⁺²(1),S^k(1/√

2)×S^n−k(1/√

2)⊂Sⁿ⁺²(1)is a submanifold with parallel mean curvature vector of normH= |^2k_n⁻ⁿ|.

Example 3.6.Denote byRP²,CP²,QP²andCayP²the projective planes over the real numbers, complex numbers, quaternions and octonions, and byψ1:RP²→S⁴(1),ψ2:CP²→S⁷(1),ψ3:QP²→S¹³(1)andψ4:CayP²→ S²⁵(1)the corresponding isometric embeddings. Letψ₁ :S²(√

3)→S⁴(1)be the isometric immersion defined by ψ₁=ψ₁◦π, whereπ:S²(√

3)→RP²is the canonical projection.

Forn2,m0, letSⁿ(1)be the great sphere inS^n+m(1)given by Sⁿ(1)=

(x1, . . . , xn+m+1)∈S^n+m(1): xn+2= · · · =xn+m+1=0 , andη_n,m:Sⁿ(1)→S^n+m(1)the inclusion. We set

φ1,p=η4,p−2◦ψ1:RP²→S^2+p(1), p2, φ_2,p=η_7,p−3◦ψ₂:CP²→S^4+p(1), p3, φ_3,p=η_13,p−5◦ψ₃:QP²→S^8+p(1), p5, φ4,p=η25,p−2◦ψ4:CayP²→S^16+p(1), p9, φ_1,p =η_4,p−2◦ψ₁:S²(√

3)→S^2+p(1), p2.

Thenφ_i,p is a minimal isometric embedding andφ_1,pis a minimal isometric immersion.

From the proof of Main Theorem 1.6 in[28]we see that, for submanifolds described in Example3.5, there exist u, v∈UxMat everyx∈Mwhich are perpendicular such thatτ (u, v)=4. Henceτ =4 onM. This implies that the conditionτ 4 of Theorem A(1)is optimal. Similarly, we haveτ ≡⁴₃ onφi,p,i=1,2,3,4, andφ_1,p described in Example3.6. So the conditionτ (x)⁴₃ in Theorem A (2) and Theorem B is optimal in the sense that there exist submanifolds which satisfyτ ≡⁴₃ but not isometric or diffeomorphic to a sphere.

Theorem A also improves the rigidity result due to Chen[10].