SPHERICAL RIGIDITIES OF

SUBMANIFOLDS IN EUCLIDEAN SPACES*

Qing-Ming Cheng

Dedicated to Professor Buchin Su for his 100th birthday

Abstract. In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space E^{n+p}. We prove that if M^{n} is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in E^{n+p} and satisfies one of the following conditions:

(1) S ≤ \frac{n^{2}H^{2}}{n−1} ,
(2) n^{2}H^{2} ≤ \frac{(n−1)R}{n−2} ,

then M^{n} is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of M^{n} and the scalar curvature of M^{n}, respectively.

On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimension and arbitrary codimension; that is, we proved that the totally umbilical sphere S^{n}(c), the totally geodesic Euclidean space E^{n}, and the generalized cylinder S^{n−1}(c) × E^{1} are only n-dimensional (n > 2) complete connected submanifolds M^{n} with constant mean curvature H in E^{n+p} if S ≤ \frac{n^{2}H^{2}}{n−1} holds.

1. Introduction

It is well known that Nash proved that every finite dimensional Riemannian manifold possesses an isometric embedding into a Euclidean space of sufficiently high dimension. Therefore, research of submanifolds in a Euclidean space E^{n+p} of n + p dimensions requires some additional conditions. In this paper, we shall agree that a submanifold means an immersed submanifold. A classical theorem of Hadamard states that a compact connected orientable hypersurface in E^{n+1} with positive sectional curvature is diffeomorphic to a standard sphere S^{n}(c). This result was generalized by Van Heijennoort [18] and Sacksteder [15]. They proved that an n-dimensional complete connected orientable hypersurface M^{n} in E^{n+1} is a boundary of a convex body in E^{n+1} if every sectional curvature of M^{n} is non-negative and at least one is positive. In particular, they proved that an ndimensional locally convex (that is, the second fundamental form is semi-definite) compact connected orientable hypersurface M^{n} in E^{n+1} is diffeomorphic to S^{n}(c). In [6] and [7], Chern and Lashof studied the total curvature of an n-dimensional

Key words and phrases: submanifolds, differentiable sphere, locally convex hypersurfaces, generalized cylinder, mean curvature, squared norm of the second fundamental form.

* Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

2000 Mathematics Subject Classification 53C42

compact connected orientable submanifold in E^{n+p}. They showed that the total curvature of an n-dimensional compact connected orientable submanifold in E^{n+p} is not less than 2c_{n+p−1}, also that, if the equality holds, then M^{n} is diffeomorphic to S^{n}(c), where c_{n+p−1} is the volume of the unit sphere S^{n+p−1}(1). Recently, using a theorem introduced by Lawson and Simons in [12], Shiohama and Xu [17] proved that an n-dimensional connected orientable complete submanifold M^{n} in E^{n+p} is homeomorphic to S^{n}(c) if n > 3 and sup_{M^{n}}(S − \frac{n^{2}H^{2}}{n−1} ) < 0. It is clear that this condition sup_{M^{n}}(S − \frac{n^{2}H^{2}}{n−1} ) < 0 yields that the mean curvature is nonzero at each point of M^{n} and M^{n} is compact by Myers theorem. In this paper, we shall prove a stronger result under a weaker condition than the one in [17]. That is, we first prove the following:

Main Theorem 1. An n-dimensional compact connected submanifold M^{n} with nonzero mean curvature H in E^{n+p} is diffeomorphic to a sphere S^{n}(c) if one of the following conditions is satisfied:

(1) S ≤ \frac{n^{2}H^{2}}{n−1} ,

(2) n^{2}H^{2} ≤ \frac{(n−1)R}{n−2},

where S and R denote the squared norm of the second fundamental form of M^{n} and the scalar curvature of M^{n}, respectively.

On the other hand, Klotz and Osserman [11] proved that a complete orientable surface M^{2} with constant mean curvature H and non-negative Gaussian curvature is isometric to a totally umbilical sphere S^{2}(c), a totally geodesic plane E^{2}, or cylinder E^{1} × S^{1}(c). It is well known that the Gaussian curvature is non-negative if and only if S ≤ \frac{n^{2}H^{2}}{n−1} holds in the case of n = 2. Next, we shall generalize the result due to Klotz and Osserman to higher dimensions and higher codimensions under the same condition of constant mean curvature.

Main Theorem 2. Let M^{n} be an n-dimensional (n > 2) complete connected submanifold with constant mean curvature H in E^{n+p}. If S ≤ \frac{n^{2}H^{2}}{n−1} is satisfied, then M is isometric to the totally umbilical sphere S^{n}(c), the totally geodesic Euclidean space E^{n}, or the generalized cylinder S^{n−1}(c) × E^{1}, where S denotes the squared norm of the second fundamental form of M^{n}.

Remark. This result due to Klotz and Osserman [11] was extended by the author and Nonaka [5] to higher dimensions and higher codimensions under the stronger condition that the mean curvature vector is parallel.

Acknowledgement. I would like to express my gratitude to Professors B. Y. Chen, K. Enomoto, K. Kenmotsu, R. Miyaoka, S. Montiel, and K. Shiohama for their valuable suggestions and discussion.

2. Preliminaries

Let E^{n+p} be an (n + p)-dimensional Euclidean space and M^{n} an n-dimensional connected submanifold in E^{n+p}. We choose a local field of orthonormal frames {e_{1}, ⋯, e_{n+p}} adapted to E^{n+p} and dual coframes {ω_{1}, ⋯, ω_{n+p}} in such a way that, restricted to the submanifold M^{n}, {e_{1}, ⋯, e_{n}} are tangent to M^{n}. Let {ω_{AB}} denote the connection forms of E^{n+p}. The canonical forms {ω_{A}} and connection forms {ω_{AB}} restricted to M^{n} are also denoted by the same symbols. We then have

(2.1)  ω_{α} = 0, α = n + 1, ⋯, n + p.

We see that e_{1}, ⋯, e_{n} is a local field of orthonormal frames adapted to the induced Riemannian metric on M^{n} and ω_{1}, ⋯, ω_{n} is a local field of its dual coframes on M^{n}. It follows from (2.1) and Cartan’s Lemma that

(2.2)  ω_{αi} = ∑_{j=1}^{n} h_{ij}^{α}ω_{j}, h_{ij}^{α} = h_{ji}^{α}.

Second fundamental form II and mean curvature vector h of M^{n} are defined by

(2.3)  II = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n} h_{ij}^{α}ω_{i}ω_{j}e_{α},

(2.4)  h = \frac{1}{n} ∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})e_{α}.

The mean curvature H of M^{n} is defined by

(2.5)  H = \frac{1}{n}\sqrt{∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α})^{2}}.

Let S = ∑_{α=n+1}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2} denote the squared norm of the second fundamental form of M^{n}. The connection form of M^{n} is characterized by the structure equations

(2.6)  dω_{i} = − ∑_{j=1}^{n} ω_{ij} ∧ ω_{j}, ω_{ij} + ω_{ji} = 0,

(2.7)  dω_{ij} = − ∑_{k=1}^{n} ω_{ik} ∧ ω_{kj} + \frac{1}{2} ∑_{k,l=1}^{n} R_{ijkl}ω_{k} ∧ ω_{l},

(2.8)  R_{ijkl} = ∑_{α=n+1}^{n+p}(h_{ik}^{α}h_{jl}^{α} − h_{il}^{α}h_{jk}^{α}).

where R_{ijkl} represents components of the curvature tensor of M^{n}. Letting R_{ij} and R denote components of the Ricci curvature and the scalar curvature of M^{n}, respectively, we obtain from (2.8):

R_{jk} = ∑_{α=n+1}^{n+p}(∑_{i=1}^{n} h_{ii}^{α}h_{jk}^{α} − ∑_{i=1}^{n} h_{ik}^{α}h_{ji}^{α}), (2.9)

R = n^{2}H^{2} − S. (2.10)

We also have

(2.11)  dω_{αβ} = − ∑_{γ=n+1}^{n+p} ω_{αγ} ∧ ω_{γβ} + \frac{1}{2} ∑_{i,j=1}^{n} R_{αβij}ω_{i} ∧ ω_{j},

(2.12)  R_{αβij} = ∑_{l=1}^{n}(h_{il}^{α}h_{lj}^{β} − h_{jl}^{α}h_{li}^{β}).

By taking the exterior differentiation of (2.2) and defining h_{ijk}^{α} by

(2.13)  ∑_{k=1}^{n} h_{ijk}^{α}ω_{k} = dh_{ij}^{α} − ∑_{k=1}^{n} h_{ik}^{α}ω_{kj} − ∑_{k=1}^{n} h_{jk}^{α}ω_{ki} − ∑_{β=n+1}^{n+p} h_{ij}^{β}ω_{βα},

we obtain Codazzi equation by straightforward computation:

(2.14)  h_{ijk}^{α} = h_{ikj}^{α} = h_{jik}^{α}.

We take the exterior differentiation of (2.13) and define h_{ijkl}^{α} by

(2.15) ∑_{l=1}^{n} h_{ijkl}^{α}ω_{l} = dh_{ijk}^{α} − ∑_{l=1}^{n} h_{ljk}^{α}ω_{li} − ∑_{l=1}^{n} h_{ilk}^{α}ω_{lj} − ∑_{l=1}^{n} h_{ijl}^{α}ω_{lk} − ∑_{β=n+1}^{n+p} h_{ijk}^{β}ω_{βα}.

Then, Ricci formula for the second fundamental form is given by

(2.16)  h_{ijkl}^{α} − h_{ijlk}^{α} = ∑_{m=1}^{n} h_{mj}^{α}R_{mikl} + ∑_{m=1}^{n} h_{im}^{α}R_{mjkl} + ∑_{β=n+1}^{n+p} h_{ij}^{β}R_{βαkl}.

The Laplacian Δh_{ij}^{α} of h_{ij}^{α} is defined by

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{ijkk}^{α}.

From the Codazzi equation (2.14) and the Ricci formula (2.16), we obtain for any α, n + 1 ≤ α ≤ n + p,

Δh_{ij}^{α} = ∑_{k=1}^{n} h_{kijk}^{α} (2.17)

= ∑_{k=1}^{n} h_{kkij}^{α} + ∑_{k,m=1}^{n} h_{km}^{α}R_{mijk}

+ ∑_{k,m=1}^{n} h_{mi}^{α}R_{mkjk} + ∑_{k=1}^{n} ∑_{β=n+1}^{n+p} h_{ki}^{β}R_{βαjk}.

The following Generalized Maximum Principle of Omori [14] and Yau [21] will be used in section 3.

Generalized Maximum Principle (Omori [14] and Yau [21]). Let M^{n} be a complete Riemannian manifold whose Ricci curvature is bounded from below and f ∈ C^{2}(M) a function bounded from above on M^{n}. Then for any ϵ > 0, there exists a point p ∈ M^{n} such that

f(p) ≥ sup f − ϵ, ∥gradf∥(p) < ϵ, Δf(p) < ϵ.

3. The reduction of codimensions

In this section, we shall prove the following:

Theorem 3.1. Let M^{n} be an n-dimensional submanifold with nonzero mean curvature H in E^{n+p} which satisfies one of the subsequent conditions. Then M^{n} lies in an (n + 1)-dimensional totally geodesic submanifold congruent to E^{n+1} of E^{n+p} if S ≤ \frac{n^{2}H^{2}}{n−1} holds:

(1) M^{n} is compact.

(2) M^{n} is complete and the mean curvature of M^{n} is constant. S denotes the squared norm of the second fundamental form of M^{n}.

Proof. Since the mean curvature of M^{n} is nonzero at each point of M^{n}, we know that e_{n+1} = \frac{h}{H} is a unit normal vector field defined globally on M^{n}. Hence, M^{n} is orientable. We define S_{1} and S_{2} as

(3.1)  S_{1} = ∑_{i,j=1}^{n}(h_{ij}^{n+1} − Hδ_{ij})^{2}, S_{2} = ∑_{α=n+2}^{n+p} ∑_{i,j=1}^{n}(h_{ij}^{α})^{2},

respectively. Then, S_{1} and S_{2} are functions defined on M^{n} globally, which do not depend on the choice of the orthonormal frame {e_{1}, ⋯, e_{n}}. Also,

(3.2)  S − nH^{2} = S_{1} + S_{2}.

From the definition of mean curvature vector h, we know that nH = ∑_{i=1}^{n} h_{ii}^{n+1} and ∑_{i=1}^{n} h_{ii}^{α} = 0 for n + 2 ≤ α ≤ n + p on M^{n}. Setting H_{α} = (h_{ij}^{α}) and defining N(A) = trace(^{t}AA) for n × n-matrix A, by making use of a direct computation we have, from (2.12) and the Gauss equation (2.8),

∑_{α=n+2}^{n+p} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{kl}^{α}R_{lijk}

= ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α})^{2} − ∑_{α=n+2}^{n+p}[trace(H_{n+1}H_{α})]^{2}

+ ∑_{α,β=n+2}^{n+p} trace(H_{α}H_{β})^{2} − ∑_{α,β=n+2}^{n+p}[trace(H_{α}H_{β})]^{2},

∑_{α=n+2}^{n+p} ∑_{i,j,k,l=1}^{n} h_{ij}^{α}h_{li}^{α}R_{lkjk}

= nH ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α}^{2})

− ∑_{α=n+2}^{n+p} trace(H_{n+1}^{2}H_{α}^{2}) − ∑_{α,β=n+2}^{n+p} trace(H_{α}H_{β}H_{β}H_{α}),

and

∑_{α,β=n+1}^{n+p} ∑_{i,j,k=1}^{n} h_{ij}^{α}h_{ki}^{β}R_{βαjk}

= ∑_{α,β=n+1}^{n+p} trace(H_{α}H_{β})^{2} − ∑_{α,β=n+1}^{n+p} trace(H_{α}H_{β}H_{β}H_{α}).

Hence, we conclude from the formula (2.17) in section 2, that

\frac{1}{2}ΔS_{2} = ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + ∑_{α=n+2}^{n+p} ∑_{i,j=1}^{n} h_{ij}^{α}Δh_{ij}^{α} (3.3)

= ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2}

+ nH ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α}^{2}) − ∑_{α=n+2}^{n+p}[trace(H_{n+1}H_{α})]^{2}

− ∑_{α,β=n+2}^{n+p} N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+2}^{n+p}[trace(H_{α}H_{β})]^{2}

+ ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α})^{2} − ∑_{α=n+2}^{n+p} trace(H_{n+1}^{2}H_{α}^{2}).

According to the following Lemma 3.1 and the definition of S_{2}, we obtain

(3.4)  − ∑_{α,β=n+2}^{n+p} N(H_{α}H_{β} − H_{β}H_{α}) − ∑_{α,β=n+2}^{n+p}[trace(H_{α}H_{β})]^{2} ≥ −\frac{3}{2}S_{2}^{2}.

Lemma 3.1 (see [13]). For symmetric matrices A_{1}, ⋯, A_{q} (q ≥ 1), put S_{αβ} = trace(A_{α}A_{β}), S_{0} = ∑_{α=1}^{q} S_{αα}, and N(A_{α}) = trace(^{t}A_{α}A_{α}). Then

∑_{α,β=1}^{q} N(A_{α}A_{β} − A_{β}A_{α}) + ∑_{α,β=1}^{q} S_{αβ}^{2} ≤ \frac{3}{2}S_{0}^{2}.

Since e_{n+1} = \frac{h}{H}, we have trace(H_{α}) = 0 for α = n + 2, ⋯, n + p and trace(H_{n+1}) = nH.

− ∑_{α=n+2}^{n+p}{trace(H_{n+1}H_{α})}^{2}+ ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α})^{2} − ∑_{α=n+2}^{n+p} trace(H_{n+1}^{2}H_{α}^{2})

= ∑_{α=n+2}^{n+p}[−{trace(H_{n+1}H_{α})}^{2} + trace(H_{n+1}H_{α})^{2} − trace(H_{n+1}^{2}H_{α}^{2})]

= ∑_{α=n+2}^{n+p}[−{trace{(H_{n+1} − HI)H_{α}}}^{2}

+ trace{(H_{n+1} − HI)H_{α}}^{2} − trace{(H_{n+1} − HI)^{2}H_{α}^{2}}], where I denotes the unit matrix.

For a fixed α, n + 2 ≤ α ≤ n + p, we can take a local orthonormal frame field {e_{1}, ⋯, e_{n}} such that h_{ji}^{α} = λ_{i}^{α}δ_{ij}. Thus, we have ∑_{i=1}^{n} λ_{i}^{α} = 0 and traceH_{α}^{2} = ∑_{i=1}^{n}(λ_{i}^{α})^{2}. Let B = H_{n+1} − HI = (b_{ij}). We have b_{ij} = b_{ji} for any i, j = 1, ⋯, n, ∑_{i=1}^{n} b_{ii} = 0 and ∑_{i,j=1}^{n} b_{ij}^{2} = S_{1}.

− [trace{(H_{n+1} − HI)H_{α}}]^{2}

+ trace{(H_{n+1} − HI)H_{α}}^{2} − trace{(H_{n+1} − HI)^{2}H_{α}^{2}}

= −{trace(BH_{α})}^{2} + trace(BH_{α})^{2} − trace(B^{2}H_{α}^{2})

= −(∑_{i=1}^{n} b_{ii}λ_{i}^{α})^{2} + ∑_{i=1}^{n} b_{ij}^{2}λ_{i}^{α}λ_{j}^{α} − ∑_{i=1}^{n} b_{ij}^{2}(λ_{i}^{α})^{2}.

Clearly, λ_{i}^{α} and b_{ij} for i, j = 1, ⋯, n satisfy the conditions in (1) of Lemma in the Appendix, which is algebraic; a proof of it can be found in [3]. For the reader’s convenience, we shall give the proof in the Appendix. We obtain

− [trace{(H_{n+1} − HI)H_{α}}]^{2}+trace{(H_{n+1} − HI)H_{α}}^{2}
− trace{(H_{n+1} − HI)^{2}H_{α}^{2}} ≥ −S_{1}traceH_{α}^{2}.

Since the two sides of the above inequality do not depend on the choice of local orthonormal frame fields, we have

∑_{α=n+2}^{n+p}[−[trace{(H_{n+1} − HI)H_{α}}]^{2} (3.5)

+ trace{(H_{n+1} − HI)H_{α}}^{2} − trace{(H_{n+1} − HI)^{2}H_{α}^{2}}]

≥ −S_{1} ∑_{α=n+2}^{n+p} traceH_{α}^{2} = −S_{1}S_{2}.

Making use of the same assertion as above, we obtain, for fixed α, n+2 ≤ α ≤ n+p,

trace{(H_{n+1} − HI)H_{α}^{2}} = ∑_{i=1}^{n} b_{ii}(λ_{i}^{α})^{2}.

From (2) and (3) of Lemma in the Appendix, we obtain

trace\{(H_{n+1} − HI)H_{α}^{2}\} ≥ −\frac{n − 2}{\sqrt{n(n − 1)}}\sqrt{S_{1}}traceH_{α}^{2}.

Hence, we conclude

nH ∑_{α=n+2}^{n+p} trace(H_{n+1}H_{α}^{2}) (3.6)

= nH ∑_{α=n+2}^{n+p} trace{(H_{n+1} − HI)H_{α}^{2}}

+ nH^{2} ∑_{α=n+2}^{n+p} traceH_{α}^{2}

= nH ∑_{α=n+2}^{n+p} trace{(H_{n+1} − HI)H_{α}^{2}} + nH^{2}S_{2}

≥ nH^{2}S_{2} − \sqrt{\frac{n}{n − 1}}(n − 2)H\sqrt{S_{1}}S_{2}.

From (3.3), (3.4), (3.5), and (3.6), we have

\frac{1}{2}ΔS_{2} ≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} (3.7)

+ (nH^{2} − \sqrt{\frac{n}{n − 1}}(n − 2)H\sqrt{S_{1}} − S_{1} − \frac{3}{2}S_{2})S_{2}

≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2}

+ (nH^{2} − \frac{n(n − 2)}{2(n − 1)}H^{2} − \frac{n − 2}{2}S_{1} − S_{1} − \frac{3}{2}S_{2})S_{2}

= ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2}

+ (nH^{2} − \frac{n(n − 2)}{2(n − 1)}H^{2} + \frac{n^{2}H^{2}}{2} − \frac{n}{2}S + \frac{(n − 3)}{2}S_{2})S_{2}

= ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + \{\frac{n}{2}(\frac{n^{2}H^{2}}{n − 1} − S) + \frac{(n − 3)}{2}S_{2}\}S_{2}

≥ ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} + \{\frac{(n − 3)}{2}S_{2}\}S_{2} ≥ 0.

When M^{n} is compact, from Stokes formula we obtain

(3.8) ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} = 0

on M^{n}; and all inequalities are equalities. Hence, we have S_{2} ≡ 0 for n > 3. When n = 3, we obtain

S_{2} ≡ 0 or S ≡ \frac{n^{2}H^{2}}{n − 1} and \sqrt{\frac{n}{n − 1}}H ≡ \sqrt{S_{1}}.

From S = S_{1} + S_{2} + nH^{2}, we also infer that S_{2} ≡ 0.

When M^{n} is complete and the mean curvature is constant, from the condition S ≤ \frac{n^{2}H^{2}}{n−1} and from (2.9) we know that the Ricci curvature of M^{n} is bounded from below. Applying the Generalized Maximum Principle of Omori [14] and Yau [21] stated in section 2 to the function S_{2}, we find that there exists a sequence {p_{k}} ⊂ M^{n} such that

(3.9)  lim_{k→∞} S_{2}(p_{k}) = sup S_{2} and lim_{k→∞} sup ΔS_{2}(p_{k}) ≤ 0.

Since S ≤ \frac{n^{2}H^{2}}{n−1} , we know that \{h_{ij}^{α}(p_{k})\}, for any i, j = 1, 2, ⋯, n and any α = n+1, ⋯, n+p, is a bounded sequence. Hence, we can assume lim_{k→∞} h_{ij}^{α}(p_{k}) = h̃_{ij}^{α}; if necessary, we can take a subsequence. From (3.7) and (3.9), by obtaining the limit of (3.7), we know that all inequalities are equalities. Hence, sup S_{2} = 0 for n > 3. When n = 3, if sup S_{2} ≠ 0, we know lim_{k→∞}(\frac{n^{2}H^{2}}{n−1} − S)(p_{k}) = 0 and

lim_{k→∞} \sqrt{\frac{n}{n−1}}H(p_{k}) = lim_{k→∞} \sqrt{S_{1}(p_{k})}. Let lim_{k→∞} H(p_{k}) = H̃, lim_{k→∞} S(p_{k})

= S̃ and lim_{k→∞} S_{1}(p_{k}) = S̃_{1}. Then, we have \frac{n^{2}H̃^{2}}{n−1} = S̃, \frac{n}{n−1} H̃^{2} = S̃_{1} and S̃ = sup S_{2} + S̃_{1} + nH̃^{2} = S̃ + sup S_{2}. This is impossible. Hence, we obtain sup S_{2} = 0. That is, S_{2} = 0 on M^{n}. From (3.7), we have

(3.10)  ∑_{α=n+2}^{n+p} ∑_{i,j,k=1}^{n}(h_{ijk}^{α})^{2} = 0

on M^{n}. Thus, we infer S_{2} ≡ 0 and (3.10) holds on M^{n} under the assumption of Theorem 3.1.

From (2.13), we have, for any α ≠ n + 1,

∑_{i,k=1}^{n} h_{iik}^{α}ω_{k} = −nHω_{αn+1}.

Hence, (3.10) yields ω_{αn+1} = 0 for any α. Thus, we know that e_{n+1} is parallel in the normal bundle T^{⊥}(M^{n}) of M^{n}. Hence, if we denote by N_{1} the normal subbundle spanned by e_{n+2}, e_{n+3}, ⋯, e_{n+p} of the normal bundle of M^{n}, then M^{n} is totally geodesic with respect to N_{1}. Since e_{n+1} is parallel in the normal bundle, we know that the normal subbundle N_{1} is invariant under parallel translation with respect to normal connection of M^{n}. Then, from Theorem 1 in [20], we conclude that M^{n} lies in a totally geodesic submanifold congruent to E^{n+1} of E^{n+p}. This completes our proof.

4. Proofs of Main Theorems

This section presents proofs of our Main Theorems.

Proof of Main Theorem 1. From Gauss equation (2.10), we have R = n^{2}H^{2} − S. Hence, we know that these two conditions in Main Theorem 1 are equivalent to each other. Thus, we shall only prove Main Theorem 1 under the condition S ≤ \frac{n^{2}H^{2}}{n−1} . From Theorem 3.1, we know that M^{n} lies in a totally geodesic submanifold E^{n+1} of E^{n+p}. We denote by H′ the mean curvature of M^{n} in E^{n+1}. Since E^{n+1} is totally geodesic in E^{n+p}, we have H = H′; that is, the mean curvature H′ of M^{n} in E^{n+1} is the same as in E^{n+p}. We also know that the squared norm S′ of the second fundamental form of M^{n} in E^{n+1} is the same as in E^{n+p}. Hence, we imply S′ ≤ \frac{n^{2}(H′)^{2}}{n−1} and H′ ≠ 0. We choose a local orthonormal frame field \{e_{1}, ⋯, e_{n}\} such that h_{ij} = λ_{i}δ_{ij} for i, j = 1, 2, ⋯, n; h_{ij} and λ_{i} denote components of the second fundamental form and principal curvatures of M^{n} in E^{n+1}, respectively. Thus, we obtain

∑_{i=1}^{n}(λ_{i})^{2} ≤ \frac{(∑_{i=1}^{n} λ_{i})^{2}}{n − 1} .

From Lemma 4.1 in Chen [1, p. 56], we have, for any i, j,

λ_{i}λ_{j} ≥ 0.

Hence, we know that the principal curvatures are non-negative on M^{n} because the mean curvature is nonzero at each point of M^{n}. Namely, M^{n} is locally convex. Therefore, M^{n} is diffeomorphic to S^{n}(c) from the result obtained by Van Heijennoort [18] and Sacksteder [15]. This completes the proof of Main Theorem 1.

Proof of Main Theorem 2. Since mean curvature H is constant, we have H = 0 or H > 0. In the case of H = 0, we have S = 0 on M^{n} since S ≤ \frac{n^{2}H^{2}}{n−1} holds. Therefore, we know that M^{n} is totally geodesic. Hence, M^{n} is isometric to the hyperplane E^{n}. Next, we assume H > 0. Thus e_{n+1} = \frac{h}{H} is a unit normal vector field defined globally on M^{n}. Hence, M^{n} is orientable. From the proof of Theorem 3.1, we know the unit normal vector field e_{n+1} = \frac{h}{H} is parallel in the normal bundle of M^{n}. Since the mean curvature is constant on M^{n}, we conclude that the mean curvature vector h = He_{n+1} is also parallel in the normal bundle of M^{n}. From results obtained by the author and Nonaka [5], we know that Main Theorem 2 is true. This completes the proof of Main Theorem 2.

5. Appendix

In the Appendix, we shall prove the following:

Lemma. (1). Let a_{1}, ⋯, a_{n} and b_{ij} for i, j = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} a_{i} = 0, ∑_{i=1}^{n} b_{ii} = 0, ∑_{i,j=1}^{n} b_{ij}^{2} = b, and b_{ij} = b_{ji} for i, j = 1, ⋯, n. Then

(5.1)  −(∑_{i=1}^{n} b_{ii}a_{i})^{2} + ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}a_{j} − ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}^{2} ≥ − ∑_{i=1}^{n} a_{i}^{2}b.

(2). Let b_{i} for i = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} b_{i} = 0 and ∑_{i=1}^{n} b_{i}^{2} = B. Then

(5.2)  ∑_{i=1}^{n} b_{i}^{4} − \frac{B^{2}}{n} ≤ \frac{(n − 2)^{2}}{n(n − 1)}B^{2}.

(3). Let a_{i} and b_{i} for i = 1, ⋯, n be real numbers satisfying ∑_{i=1}^{n} a_{i} = 0 and ∑_{i=1}^{n} a_{i}^{2} = a. Then

(5.3)  ∑_{i=1}^{n} a_{i}b_{i}^{2} ≥ −\sqrt{∑_{i=1}^{n} b_{i}^{4} − \frac{(∑_{i=1}^{n} b_{i}^{2})^{2}}{n}} \sqrt{a}.

Proof. In order to prove (1), we consider the function

(5.4)  f(x_{ij}) = −(∑_{i=1}^{n} x_{ii}a_{i})^{2} − \frac{1}{2} ∑_{i,j=1}^{n} x_{ij}^{2}(a_{j} − a_{i})^{2},

subject to the constraint conditions

(5.5)  ∑_{i=1}^{n} x_{ii} = 0 and ∑_{i,j=1}^{n} x_{ij}^{2} = b.

Making use of Lagrangian multipliers, we shall calculate the minimum of the function f(x_{ij}) with constraint condition (5.5). Let

g = f(x_{ij}) + λ ∑_{i=1}^{n} x_{ii} + μ( ∑_{i,j=1}^{n} x_{ij}^{2} − b),

where λ and μ are the Lagrangian multipliers. We have

g = −(∑_{i=1}^{n} a_{i}x_{ii})^{2} − \frac{1}{2} ∑_{i,j=1}^{n} x_{ij}^{2}(a_{j} − a_{i})^{2} + λ ∑_{i=1}^{n} x_{ii} + μ( ∑_{i,j=1}^{n} x_{ij}^{2} − b).

If f attains its minimum f_{0} at some point (x_{ij}), we have

− 2 ∑_{i=1}^{n} a_{i}x_{ii}a_{j} + λ + 2μx_{jj} = 0, for j = 1, ⋯, n, (5.6)

− x_{ij}(a_{j} − a_{i})^{2} + 2μx_{ij} = 0, for i ≠ j. (5.7)

Hence,

−(∑_{i=1}^{n} a_{i}x_{ii})^{2} + μ ∑_{j=1}^{n} x_{jj}^{2} = 0,

−\frac{1}{2} ∑_{i,j=1}^{n} x_{ij}^{2}(a_{j} − a_{i})^{2} + μ ∑_{i,j=1,i≠j}^{n} x_{ij}^{2} = 0.

Thus,

f_{0} = −μb.

From (5.6) and ∑_{i=1}^{n} a_{i} = 0, we obtain λ = 0 and

(μ − ∑_{j=1}^{n} a_{j}^{2}) ∑_{i=1}^{n} x_{ii}a_{i} = 0,

μ ∑_{j=1}^{n} x_{jj}^{2} − (∑_{i=1}^{n} x_{ii}a_{i})^{2} = 0.

If ∑_{i=1}^{n} x_{ii}a_{i} ≠ 0, we have μ = ∑_{j=1}^{n} a_{j}^{2}. Hence,

f_{0} = −μb = − ∑_{j=1}^{n} a_{j}^{2}b.

If ∑_{i=1}^{n} x_{ii}a_{i} = 0, we have μ ∑_{j=1}^{n} x_{jj}^{2} = 0. μ = 0 yields f_{0} = 0. If μ ≠ 0, we have ∑_{j=1}^{n} x_{jj}^{2} = 0. Hence, b = 0 or there exists i ≠ j such that x_{ij} ≠ 0. From (5.7), we obtain

2μ = (a_{i} − a_{j})^{2} ≤ 2 ∑_{j=1}^{n} a_{j}^{2}.

Therefore,

f_{0} ≥ − ∑_{j=1}^{n} a_{j}^{2}b.

Since ∑_{i=1}^{n} a_{i} = 0, ∑_{i=1}^{n} b_{ii} = 0, ∑_{i,j=1}^{n} b_{ij}^{2} = b, and b_{ij} = b_{ji} for i, j = 1, ⋯, n hold, we have

− (∑_{i=1}^{n} b_{ii}a_{i})^{2} + ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}a_{j} − ∑_{i,j=1}^{n} b_{ij}^{2}a_{i}^{2}

= −(∑_{i=1}^{n} b_{ii}a_{i})^{2} − \frac{1}{2} ∑_{i,j=1}^{n} b_{ij}^{2}(a_{j} − a_{i})^{2} ≥ − ∑_{j=1}^{n} a_{j}^{2}b.

Thus, we complete the proof of (1) Lemma.

For the proof of (2), we consider the function

f(y) = ∑_{i=1}^{n} y_{i}^{4} − \frac{B^{2}}{n}

with constraint conditions ∑_{i=1}^{n} y_{i} = 0 and ∑_{i=1}^{n} y_{i}^{2} = B.

Since ∑_{i=1}^{n} y_{i}^{2} = B, we know that at least one of the y_{i}^{2}’s is not less than \frac{B}{n}. We assume the y_{n}^{2} ≥ \frac{B}{n}, without loss of generality. From ∑_{i=1}^{n} y_{i} = 0, we have

y_{n}^{2} = (∑_{i=1}^{n−1} y_{i})^{2} ≤ (n − 1) ∑_{i=1}^{n−1} y_{i}^{2} = (n − 1)(B − y_{n}^{2}),

y_{n}^{2} − \frac{B}{2} = ∑_{1≤i<j≤n−1} y_{i}y_{j},

y_{n}^{2} ≤ \frac{(n − 1)B}{n} .

Hence,

f(y) = ∑_{i=1}^{n−1} y_{i}^{4} + y_{n}^{4} − \frac{B^{2}}{n}

= (∑_{i=1}^{n−1} y_{i}^{2})^{2} − 2 ∑_{1≤i<j≤n−1} y_{i}^{2}y_{j}^{2} + y_{n}^{4} − \frac{B^{2}}{n}

≤ (B − y_{n}^{2})^{2} − \frac{4}{(n − 1)(n − 2)}( ∑_{1≤i<j≤n−1} y_{i}y_{j})^{2} + y_{n}^{4} − \frac{B^{2}}{n}

= \frac{2n(n − 3)}{(n − 1)(n − 2)}(y_{n}^{4} − By_{n}^{2}) + (\frac{n − 1}{n} − \frac{1}{(n − 1)(n − 2)})B^{2}.

Since the maximum of the function t^{2} − Bt in the interval [\frac{1}{n}B, \frac{n−1}{n}B] is −\frac{n−1}{n^{2}}B^{2}, we obtain

f(y) ≤ \frac{(n − 2)^{2}}{n(n − 1)}B^{2}.

This completes the proof of (2) of Lemma.

Making use of the Lagrangian multipliers, we calculate the minimum of the function g(x) = ∑_{i=1}^{n} x_{i}b_{i}^{2} with constraint conditions ∑_{i=1}^{n} x_{i} = 0 and ∑_{i=1}^{n} x_{i}^{2} = a. If the function g(x) attains its minimum g_{0} at some point x, then we have, at x,

b_{i}^{2} + λ + 2μx_{i} = 0, for i = 1, ⋯, n,

where λ and μ are the Lagrangian multipliers. Hence, we have

g_{0} = −2μa, λ = −\frac{∑_{i=1}^{n} b_{i}^{2}}{n} ,

∑_{i=1}^{n} b_{i}^{4} − \frac{(∑_{i=1}^{n} b_{i}^{2})^{2}}{n} + 2μg_{0} = 0.

Thus, (3) of Lemma is true.

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Department of Mathematics, Faculty of Science and Engineering Saga University, Saga 840-8502, Japan cheng@ms.saga-u.ac.jp