Mean Curvature and Shape Operator of Slant Immersions in a Sasakian Space Form

Muck Main Tripathi, Jean-Sic Kim and Son-Be Kim

Abstract

For submanifolds, in a Sasakian space form, which are tangential to the structure vector field, we establish a basic inequality between squared mean curvature and Ricci curvature. Equality cases are also discussed. Some applications of these results are given for slant, invariant, anti-invariant and CR-submanifolds. We also establish an inequality between the shape operator and the sectional curvature for slant submanifolds in a Sasakian space form. In particular, we give similar results for invariant and anti-invariant submanifolds.

Mathematics Subject Classification: 53C40, 53C25. Key words: Sasakian space form, invariant submanifold, anti-invariant submanifold, slant submanifold, CR-submanifold, totally geodesic submanifold, Ricci curvature, sectional curvature and squared mean curvature.

1 Introduction

According to B.-Y. Chen, one of the basic problems in submanifold theory is to find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. In [5], he establishes a relationship between sectional curvature function K and the shape operator for submanifolds in real space forms. In [6], he also gives a relationship between Ricci curvature and squared mean curvature.

A contact version of B.-Y. Chen’s inequality and its applications to slant immersions in a Sasakian space form M̃ (c) are given in [4]. In the present paper, we continue the study of submanifolds in a Sasakian space form, which are tangent to the structure vector field. Necessary details about Sasakian space forms and slant submanifolds are reviewed in section 2. In section 3, for those submanifolds in Sasakian space forms which are tangential to the structure vector field, we establish a basic inequality between Ricci curvature and squared mean curvature function. We also discuss equality cases. As applications, we state similar results for slant, invariant, anti-invariant and CR-submanifolds. In the last section, we establish an inequality between the shape operator and the sectional curvature for slant submanifolds in a Sasakian space form. In particular, we give similar results for invariant and anti-invariant submanifolds.

Balkan Journal of Geometry and Its Applications, Vol.7, No.1, 2002, pp. 101-111.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

2 Preliminaries

Let M̃ be a (2m + 1)-dimensional almost contact manifold endowed with an almost contact structure (φ, ξ, η), that is, φ is a (1, 1) tensor field, ξ is a vector field and η is 1-form such that φ^{2} = −I + η ⊗ ξ and η (ξ) = 1. Then, φ (ξ) = 0 and η ∘ φ = 0. The almost contact structure is said to be normal if in the product manifold M̃ × R the induced almost complex structure J defined by J (X, λd/dt) = (φX − λξ, η (X) d/dt) is integrable, where X is tangent to M̃, t is the coordinate of R and λ is a smooth function on M̃ × R. The condition for almost contact structure to be normal is equivalent to vanishing of the torsion tensor [φ, φ] + 2dη ⊗ ξ, where [φ, φ] is the Nijenhuis tensor of φ.

Let g be a compatible Riemannian metric with the structure (φ, ξ, η), that is, g (φX, φY) = g (X, Y) − η (X) η (Y) or equivalently, g (X, φY) = −g (φX, Y) and g (X, ξ) = η (X) for all X, Y ∈ T M̃. Then, M̃ becomes an almost contact metric manifold equipped with the almost contact metric structure (φ, ξ, η, g). Moreover, if g (X, φY) = dη (X, Y), then M̃ is said to have a contact metric structure (φ, ξ, η, g), and M̃ is called a contact metric manifold. A normal contact metric structure in M̃ is a Sasakian structure and M̃ is a Sasakian manifold. A necessary and sufficient condition for an almost contact metric structure to be a Sasakian structure is

( ˜∇_{X}φ) Y = g (X, Y) ξ − η (Y) X, X, Y ∈ T M̃, (1)

where ˜∇ is the Levi-Civita connection of the Riemannian metric g. R^{2m+1} and S^{2m+1} are equipped with standard Sasakian structures. For more details, we refer to [2].

The sectional curvature K̃ (X ∧ φX) of a plane section spanned by a unit vector X orthogonal to ξ is called a φ-sectional curvature. If M̃ has constant φ-sectional curvature c then it is called a Sasakian space form and is denoted by M̃ (c). The curvature tensor R̃ of a Sasakian space form M̃ (c) is given by

4R̃ (X, Y) Z = (c + 3) {g (Y, Z) X − g (X, Z) Y } +
+ (c − 1) {g (φY, Z) φX − g (φX, Z) φY −
(2)  − 2g (φX, Y) φZ + η (X) η (Z) Y −
− η (Y) η (Z) X + g (X, Z) η (Y) ξ − g (Y, Z) η (X) ξ} .

Let M be an (n + 1)-dimensional submanifold immersed in an almost contact metric manifold M̃(φ, ξ, η, g). Let g denote the induced metric on M also. We denote by σ the second fundamental form of M and by A_{N} the shape operator associated to any vector N in the normal bundle T^{⊥}M. Then g (σ (X, Y) , N) = g (A_{N}X, Y) for all X, Y ∈ T M and N ∈ T^{⊥}M. The Gauss equation is

(3)  R̃ (X, Y, Z, W) = R (X, Y, Z, W) − g (σ (X, W) , σ (Y, Z))
+ g (σ (X, Z) , σ (Y,W))

for all X, Y, Z, W ∈ T M, where R is the induced curvature tensor of M. The relative null space of M at a point p ∈ M is defined by

N_{p} = {X ∈ T_{p}M| σ(X, Y) = 0, for all Y ∈ T_{p}M} .

Let {e_{1}, ..., e_{n+1}} be an orthonormal basis of the tangent space T_{p}M. The mean curvature vector H (p) at p ∈ M is

H (p) ≡ \frac{1}{n + 1} ∑_{i=1}^{n+1} σ (e_{i}, e_{i}) . (4)

The submanifold M is totally geodesic in M̃ if σ = 0; minimal if H = 0; and totally umbilical if σ (X, Y) = g (X, Y) H for all X, Y ∈ T M. We put

σ_{ij}^{r} = g (σ (e_{i}, e_{j}) , e_{r}) and ∥σ∥^{2} = ∑_{i,j=1}^{n+1} g (σ (e_{i}, e_{j}) , σ (e_{i}, e_{j})) ,

where e_{r} belongs to an orthonormal basis {e_{n+2}, ..., e_{2m+1}} of the normal space T_{p}^{⊥}M. The scalar curvature τ (p) at p ∈ M is given by

τ (p) = ∑_{i<j} K (e_{i} ∧ e_{j}) , (5)

where K (e_{i} ∧ e_{j}) is the sectional curvature of the plane section spanned by e_{i} and e_{j}. For a vector field X in M, we put

φX = P X + F X, P X ∈ T M, F X ∈ T^{⊥}M.

Thus, P is an endomorphism of the tangent bundle of M and satisfies g (X, P Y) = −g (P X, Y) for all X, Y ∈ T M. The squared norm of P is given by

∥P ∥^{2} = ∑_{i,j=1}^{n+1} g (e_{i}, P e_{j})^{2}

for any local orthonormal basis {e_{1}, e_{2}, …, e_{n+1}} for T_{p}M.

A submanifold M of an almost contact metric manifold with ξ ∈ T M is called a semi-invariant submanifold ([1]) or a contact CR submanifold ([8]) if there exists two differentiable distributions D and D^{⊥} on M such that (i) T M = D ⊕ D^{⊥} ⊕ E, (ii) the distribution D is invariant by φ, i.e., φ(D) = D, and (iii) the distribution D^{⊥} is anti-invariant by φ, i.e., φ(D^{⊥}) ⊆ T^{⊥}M.

The submanifold M tangent to ξ is said to be invariant or anti-invariant ([8]) according as F = 0 or P = 0. Thus, a CR-submanifold is invariant or anti-invariant according as D^{⊥} = {0} or D = {0}. A proper CR-submanifold is neither invariant nor anti-invariant.

For each non zero vector X ∈ T_{p}M, such that X is not proportional to ξ_{p}, we denote the angle between φX and T_{p}M by θ (X). Then M is said to be slant ([7],[3]) if the angle θ (X) is constant, that is, it is independent of the choice of p ∈ M and X ∈ T_{p}M − {ξ}. The angle θ of a slant immersion is called the slant angle of the immersion. Invariant and anti-invariant immersions are slant immersions with slant angle θ = 0 and θ = π/2 respectively. A proper slant immersion is neither invariant nor anti-invariant.

3 Mean curvature and Ricci curvature

Let M be an (n + 1)-dimensional submanifold in a (2m + 1)-dimensional Sasakian space form M̃(c) tangential to the structure vector field ξ. In view of (2) and (3), it implies that

R (X, Y, Z, W) = \frac{c + 3}{4} \{g (X, W) g (Y, Z) − g (X, Z) g (Y, W)\} +
+ \frac{c − 1}{4} \{g (X, φW) g (Y, φZ) − g (X, φZ) g (Y, φW) −
(6)  − 2g (X, φY) g (Z, φW) −
− g (X, W) η (Y) η (Z) + g (X, Z) η (Y) η (W) −
− g (Y, Z) η (X) η (W) + g (Y, W) η (X) η (Z)} +
+ g (σ (X, W) , σ (Y, Z)) − g (σ (X, Z) , σ (Y, W))

for all X, Y, Z, W ∈ T M, where R is the induced curvature tensor of M. Thus, we have

(n + 1)^{2} ∥H∥^{2} = 2τ + ∥σ∥^{2} − \frac{1}{4}n (n + 1) (c + 3) − \frac{1}{4} (3 ∥P ∥^{2} − 2n) (c − 1) . (7)

In [6], B.-Y. Chen established a relationship between Ricci curvature and the squared mean curvature for a submanifold in a real space form as follows.

Theorem 3.1 Let M be an n-dimensional submanifold in a real space form R^{m} (c). Then,

1. For each unit vector X ∈ T_{p}M, we have

∥H∥^{2} ≥ \frac{4}{n^{2}} \{Ric (X) − (n − 1) c\} . (8)

2. If H(p) = 0, then a unit vector X ∈ T_{p}M satisfies the equality case of (8) if and only if X lies in the relative null space N_{p} at p.

3. The equality case of (8) holds for all unit vectors X ∈ T_{p}M, if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.

In this section, we find similar results for different kind of submanifolds in a Sasakian space form.

Theorem 3.2 Let M be an (n + 1)-dimensional submanifold in a (2m+1)-dimensional Sasakian space form M̃(c) tangential to the structure vector field ξ. Then,

(i) For each unit vector U ∈ T_{p}M, we have

4Ric (U)≤(n + 1)^{2} ∥H∥^{2}+n (c+3)+{3 ∥P U∥^{2}−(n−1) η (U)^{2}− 1} (c − 1) . (9)

(ii) If H(p) = 0, a unit vector U ∈ T_{p}M satisfies the equality case of (9) if and only if U belongs to the relative null space N_{p}. (iii) The equality case of (9) holds for all unit vectors U ∈ T_{p}M if and only if M is a totally geodesic submanifold.

Proof. We choose an orthonormal basis {e_{1}, ..., e_{n+1}, e_{n+2}, ..., e_{2m+1}} such that e_{1}, ..., e_{n+1} ∈ T_{p}M. The squared second fundamental form and the squared mean curvature vector also satisfy

∥σ∥^{2} = \frac{1}{2} (n + 1)^{2} ∥H∥^{2} + \frac{1}{2} ∑_{r=n+2}^{2m+1}(σ_{11}^{r} − σ_{22}^{r} − ⋯ − σ_{n+1 n+1}^{r})^{2} +

(10)  + 2 ∑_{r=n+2}^{2m+1} ∑_{j=2}(σ_{1j}^{r})^{2} − 2 ∑_{r=n+2}^{2m+1} ∑_{2≤i<j≤n+1} (σ_{ii}^{r}σ_{jj}^{r} − (σ_{ij}^{r})^{2}) .

From (7) and (10), we get

\frac{1}{4} (n + 1)^{2} ∥H∥^{2} = τ − \frac{1}{8}n(n + 1) (c + 3) − \frac{1}{8} (3 ∥P ∥^{2} − 2n) (c − 1) +
(11)  + \frac{1}{4} ∑_{r=n+2}^{2m+1}(σ_{11}^{r} − σ_{22}^{r} − ⋯ − σ_{n+1 n+1}^{r})^{2} + ∑_{r=n+2}^{2m+1} ∑_{j=2}^{n+1}(σ_{1j}^{r})^{2} −

− ∑_{r=n+2}^{2m+1} ∑_{2≤i<j≤n+1} (σ_{ii}^{r}σ_{jj}^{r} − (σ_{ij}^{r})^{2}) .

From the equation of Gauss we also have

K (e_{i} ∧ e_{j}) = ∑_{r=n+2}^{2m+1} (σ_{ii}^{r}σ_{jj}^{r} − (σ_{ij}^{r})^{2}) + \frac{c + 3}{4}+

+ \frac{c − 1}{4} (3g (e_{i}, P e_{j})^{2} − η (e_{i})^{2} − η (e_{j})^{2}) ,

which gives

∑_{2≤i<j≤n+1}K(e_{i}∧e_{j}) = ∑_{r=n+2}^{2m} ∑_{2≤i<j≤n+1}(σ_{ii}^{r}σ_{jj}^{r} − (σ_{ij}^{r})^{2}) + \frac{1}{8}n(n − 1) (c + 3) +

(12)  + \frac{1}{8}\{3 ∥P ∥^{2}−6 ∥P e_{1}∥^{2}−2(n−1)(1−η(e_{1})^{2}\}(c−1).

From (11) and (12), we get

\frac{1}{4} (n + 1)^{2} ∥H∥^{2} = τ − ∑_{2≤i<j≤n+1} K (e_{i} ∧ e_{j}) −

− \frac{1}{4}n (c + 3) − \frac{1}{4} (3 ∥P e_{1}∥^{2} − (n − 1) η (e_{1})^{2} − 1) (c − 1) +
+ \frac{1}{4} ∑_{r=n+2}^{2m+1}(σ_{11}^{r} − σ_{22}^{r} − ⋯ − σ_{n+1 n+1}^{r})^{2} + ∑_{r=n+2}^{2m+1} ∑_{j=2}(σ_{1j}^{r})^{2}.

or

Ric (e_{1}) = \frac{1}{4} \{(n + 1)^{2} ∥H∥^{2} + n (c + 3) +

(13)  + (3 ∥P e_{1}∥^{2} − (n − 1) η (e_{1})^{2} − 1) (c − 1)} −
− \frac{1}{4} ∑_{r=n+2}^{2m+1}(σ_{11}^{r} − σ_{22}^{r} − ⋯ − σ_{n+1 n+1}^{r})^{2} − ∑_{r=n+2}^{2m+1} ∑_{j=2}(σ_{1j}^{r})^{2}.

Since e_{1} = X can be chosen to be any arbitrary unit vector in T_{p}M, the above equation implies (9).

In view of (13), the equality case of (9) is valid if and only if

σ_{11}^{r} = σ_{22}^{r} + ⋯ + σ_{n+1 n+1}^{r},
(14)  σ_{12}^{r} = ⋯ = σ_{1 n+1}^{r} = 0, r ∈ {n + 2, …, 2m + 1} .

If H(p) = 0, (14) implies that e_{1} = X belongs to the relative null space N_{p} at p. Conversely, if e_{1} = X lies in the relative null space, then (14) holds because H(p) = 0 is assumed. This proves statement (ii).

Now, we prove (iii). Assume that the equality case of (9) for all unit tangent vectors to M at p ∈ M is true. Then, in view of (13), for each r ∈ {n + 2, …, 2m + 1}, we have

2σ_{ii}^{r} = σ_{11}^{r} + ⋯ + σ_{n+1 n+1}^{r}, i ∈ {1, ..., n + 1} ,
(15)  σ_{ij}^{r} = 0, i ≠ j.

Thus, we have two cases, namely either n = 1 or n ≠ 1. In the first case p is a totally umbilical point, while in the second case p is a totally geodesic point. Since ξ ∈ T M, therefore each totally umbilical point is totally geodesic. Thus in both the cases, p is a totally geodesic point. The proof of converse part is straightforward. □

The above theorem implies the following three results for slant, invariant and anti-invariant submanifolds isometrically immersed in a Sasakian space form.

Theorem 3.3 Let M be an (n + 1)-dimensional θ-slant submanifold isometrically immersed in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. Then

(i) For each unit vector U ∈ T_{p}M, we have

4Ric (U) ≤ (n + 1)^{2} ∥H∥^{2} + n (c + 3)

+ {3 cos^{2} θ − (n − 1 + 3 cos^{2} θ) η (U)^{2} − 1} (c − 1) . (16)

(ii) If H(p) = 0, a unit vector U ∈ T_{p}M satisfies the equality case of (16) if and only if U ∈ N_{p}. (iii) The equality case of (16) holds for all unit vectors U ∈ T_{p}M if and only if M is a totally geodesic submanifold.

Proof. A θ-slant submanifold M of an almost contact metric manifold satisfies

g (P X, P Y) = cos^{2} θg (φX, φY) , g (F X, F Y) = sin^{2} θg (φX, φY) (17)

for all X, Y ∈ T M. In view of (17), for a unit vector U ∈ T_{p}M, we get

∥P U∥^{2} = g (P U, P U) = cos^{2} θ (1 − η (U)^{2}) .

Using this in (9), we get (16). Rest of the proof is similar to that of Theorem 3.2. □

Theorem 3.4 Let M be an (n + 1)-dimensional invariant submanifold isometrically immersed in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. Then,

(i) For each unit vector U ∈ T_{p}M, we have

4Ric (U) ≤ (n + 1)^{2} ∥H∥^{2} + n (c + 3) + {2 − (n + 2) η (U)^{2}} (c − 1) . (18)

(ii) If H(p) = 0, a unit vector U ∈ T_{p}M satisfies the equality case of (18) if and only if U ∈ N_{p}. (iii) The equality case of (18) holds for all unit vectors U ∈ T_{p}M if and only if M is a totally geodesic submanifold.

Theorem 3.5 Let M be an (n + 1)-dimensional anti-invariant submanifold isometrically immersed in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. Then,

(i) For each unit vector U ∈ T_{p}M, we have

4Ric (U) ≤ (n + 1)^{2} ∥H∥^{2} + n (c + 3) − {(n − 1) η (U)^{2} + 1} (c − 1) . (19)

(ii) If H(p) = 0, a unit vector U ∈ T_{p}M satisfies the equality case of (19) if and only if U ∈ N_{p}. (iii) The equality case of (19) holds for all unit vectors U ∈ T_{p}M if and only if M is a totally geodesic submanifold.

We also have the following

Theorem 3.6 Let M be an (n + 1)-dimensional CR-submanifold in a Sasakian space form M̃ (c). Then, the following statements are true.

1. For each unit vector U ∈ D, we have

4Ric (U) ≤ (n + 1)^{2} ∥H∥^{2} + (n + 2) c + 3n − 2. (20)

2. For each unit vector U ∈ D^{⊥}, we have

4Ric (U) ≤ (n + 1)^{2} ∥H∥^{2} + (n − 1) c + 3n + 1. (21)

3. If H(p) = 0, a unit vector U ∈ D (resp. D^{⊥}) satisfies the equality case of (20) (resp. (21)) if and only if U ∈ N_{p}.

4 Shape operator for slant immersion

Let M be an (n + 1)-dimensional θ-slant submanifold in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. Let p ∈ M and a number

b > \frac{c + 3}{4} + \frac{3 (c − 1)}{4} cos^{2} θ

such that the sectional curvature K ≥ b at p. We choose an orthonormal basis {e_{1}, …, e_{n+1} = ξ, e_{n+2}, …, e_{2m+1}} at p such that e_{n+2} is parallel to the mean curvature vector H, and e_{1}, …, e_{n+1} diagonalize the shape operator A_{n+2}. Then we have

A_{n+2} = ( \begin{matrix}a_{1} & 0 & ⋯ & 0 \\ 0 & a_{2} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{n+1} \end{matrix} ) , (22)

A_{r} = (σ_{ij}^{r}) , trace A_{r} = ∑_{i=1}^{n+1} σ_{ii}^{r} = 0, i, j = 1, ..., n + 1; r = n + 3, ..., 2m + 1. (23)

For i ≠ j, we put u_{ij} ≡ a_{i}a_{j} = u_{ji}. (24)

In view of Gauss equation (6), for X = Z = e_{i}, Y = W = e_{j}, we have

u_{ij} ≥ b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4}g (e_{i}, P e_{j})^{2} − ∑_{r=n+3}^{2m+1} (σ_{ii}^{r}σ_{jj}^{r} − (σ_{ij}^{r})^{2}) . (25)

Now, we prove the following Lemma.

Lemma 4.1 For u_{ij} we have

(1) For any fixed i ∈ {1, ..., n + 1}, we find

∑_{i≠j} u_{ij} ≥ n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) .

(2) For distinct i, j, k ∈ {1, ..., n + 1} it follows that a_{i}^{2} = u_{ij}u_{ik}/u_{jk}.

(3) For a fixed k, 1 ≤ k ≤ [\frac{n + 1}{2}] and for each B ∈ S_{k} ≡ \{B ⊂ \{1, ..., n + 1\} : |B| =

k}, we have

∑_{j∈B} ∑_{t∈B̄} u_{jt} ≥ k (n − k + 1) (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) ,

where B̄ is the complement of B in {1, ..., n + 1}.

(4) For distinct i, j ∈ {1, ..., n + 1}, it follows that u_{ij} > 0.

Proof. (1) From (23), (24) and (25), we obtain

∑_{i≠j} u_{ij} ≥ n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) − ∑_{r=n+3}^{2m+1} (σ_{ii}^{r} (∑_{j≠i} σ_{jj}^{r}) − ∑_{j≠i} (σ_{ij}^{r})^{2}) =

= n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) − ∑_{r=n+3}^{2m+1} (σ_{ii}^{r} (−σ_{ii}^{r}) − ∑_{j≠i} (σ_{ij}^{r})^{2}) =

= n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) + ∑_{r=n+3}^{2m+1} ∑_{j=1}^{n+1} (σ_{ij}^{r})^{2} ≥

≥ n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) > 0.

(2) We have u_{ij}u_{ik}/u_{jk} = a_{i}a_{j}a_{i}a_{k}/a_{j}a_{k} = a_{i}^{2}.

(3) Let B = {1, ..., k} and B̄ = {k + 1, ..., n + 1}. Then

∑_{j∈B} ∑_{t∈B̄} u_{jt} ≥ k (n − k + 1) (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) −

− ∑_{r=n+3}^{2m+1} (∑_{j=1}^{k} ∑_{t=k+1}^{n+1} [σ_{jj}^{r}σ_{tt}^{r} − (σ_{jt}^{r})^{2}]) =

= k (n − k + 1) (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) +

+ ∑_{r=n+3}^{2m+1} (∑_{j=1}^{k} ∑_{t=k+1}^{n+1} (σ_{jt}^{r})^{2} + ∑_{j=1}^{k} (σ_{jj}^{r})) ≥

≥ k (n − k + 1) (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) .

(4) For i ≠ j, if u_{ij} = 0 then a_{i} = 0 or a_{j} = 0. The statement a_{i} = 0 implies that u_{il} = a_{i}a_{l} = 0 for all l ∈ {1, ..., n + 1}, l ≠ i. Then, we get

∑_{j≠i} u_{ij} = 0,

which is a contradiction with (1). Thus, for i ≠ j, it follows that u_{ij} ≠ 0. We assume that u_{1 n+1} < 0. From (2), for 1 < i < n + 1, we get u_{1i}u_{i n+1} < 0. Without loss of generality, we may assume

\begin{matrix}u_{12}, …, u_{1l}, , u_{l+1 n+1}, …, u_{n n+1} > 0, \\ u_{1 l+1}, …, u_{1 n+1}, u_{2 n+1}, …, u_{l n+1} < 0, \end{matrix} (26)

for some [\frac{n}{2} + 1] ≤ l ≤ n. If l = n, then u_{1 n+1} + u_{2 n+1} + ⋯ + u_{n n+1} < 0, which contradicts to (1). Thus, l < n. From (2), we get:

a_{n+1}^{2} = \frac{u_{i n+1}u_{t n+1}}{u_{i t}} > 0, (27)

where 2 ≤ i ≤ l, l + 1 ≤ t ≤ n. By (26) and (27), we obtain u_{it} < 0, which implies that

∑_{i=1}^{l} ∑_{t=l+1}^{n+1} u_{it} = ∑_{i=2}^{l} ∑_{t=l+1}^{n} u_{it} + ∑_{i=1}^{l} u_{i n+1} + ∑_{t=l+1}^{n+1} u_{1t} < 0,

which is a contradiction to (3). Thus (4) is proved. □

B.-Y. Chen establishes a sharp relationship between the shape operator and the sectional curvature for submanifolds in real space forms [5]. In the following theorem, we establish a similar inequality between the shape operator and the sectional curvature for slant submanifolds in a Sasakian space form.

Theorem 4.2 Let M be an (n + 1)-dimensional slant submanifold isometrically immersed in a (2m+1)-dimensional Sasakian space form M̃(c). If at a point p ∈ M there exists a number b > (c + 3) /4 + (3/4) (c − 1) cos^{2} θ such that the sectional curvature K ≥ b at p, then the shape operator A_{H} at the mean curvature vector satisfies

A_{H} > \frac{n}{n + 1} (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) I_{n}, at p, (28)

where I_{n} is the identity map.

Proof. Let p ∈ M and a number b > (c + 3) /4 + (3/4) (c − 1) cos^{2} θ such that the sectional curvature K ≥ b at p. We choose an orthonormal basis {e_{1}, …, e_{n+1}, e_{n+2}, …, e_{2m+1}} at p such that e_{n+2} is parallel to the mean curvature vector H, and e_{1}, …, e_{n+1} diagonalize the shape operator A_{n+2}. Now, from Lemma 4.1 it follows that a_{1}, ..., a_{n+1} have the same sign. We assume that a_{j} > 0 for all j ∈ {1, …, n + 1}. Then

∑_{j≠i} u_{ij} = a_{i} (a_{1} + ⋯ + a_{n+1}) − a_{i}^{2} ≥ n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) . (29)

From (29) and (22), we obtain

a_{i} (n + 1) ∥H∥ ≥ n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) + a_{i}^{2}
> n (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) ,

which implies that

a_{i} ∥H∥ > \frac{n}{n + 1} (b − \frac{c + 3}{4} − \frac{3 (c − 1)}{4} cos^{2} θ) .

Hence, we get (28). □

In particular, the above theorem implies the following two theorems.

Theorem 4.3 Let M be an (n + 1)-dimensional anti-invariant submanifold isometrically immersed in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. If at a point p ∈ M there exists a number b > (c + 3) /4 such that the sectional curvature K ≥ b at p, then the shape operator A_{H} at the mean curvature vector satisfies

A_{H} > \frac{n}{n + 1} (b − \frac{c + 3}{4}) I_{n}, at p. (30)

Theorem 4.4 Let M be an (n + 1)-dimensional invariant submanifold isometrically immersed in a (2m + 1)-dimensional Sasakian space form M̃(c) such that ξ ∈ T M. If at a point p ∈ M there exists a number b > c such that the sectional curvature K ≥ b at p, then the shape operator A_{H} at the mean curvature vector satisfies

A_{H} > \frac{n}{n + 1} (b − c) I_{n}, at p. (31)

The above equation is same as equation (5.1) of Theorem 4.1 in the paper of B.-Y. Chen [5]. Acknowledgement. Postdoctoral Researcher at Chonnam National University, Korea in Brain Korea 21. Coordinator Project: Professor Minkyu Kwak.

References

[1] A. Bejancu, Geometry of CR-submanifolds, Mathematics and Its Applications (East European Series), 23. D. Reidel Publishing Co., Dordrecht, 1986.

[2] D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhauser Boston, Inc., Boston, MA, 2002.

[3] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42 (2000), no. 1, 125–138.

[4] A. Carriazo, A contact version of B.-Y. Chen’s inequality and its applications to slant immersions, Kyungpook Math. J. 39 (1999), no. 2, 465-476.

[5] B.-Y. Chen, Mean curvature and shape operator of isometric immersions in real space form, Glasgow Math. J. 38 (1996), 87-97.

[6] B.-Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 41 (1999), 33-41.

[7] A. Lotta, Slant submanifolds in contact geometry, Bull. Math. Soc. Roumanie 39 (1996), 183-186.

[8] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, 3. World Scientific. 1984.

Mukut Mani Tripathi Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 007, India Present Address: Department of Mathematics, Chonnam National University, Kwangju 500-757, Korea email: mm tripathi@hotmail.com

Jeong-Sik Kim Department of Mathematics Education Sunchon National University, Sunchon 540-742, Korea email: jskim01@hanmir.com

Seon-Bu Kim Department of Mathematics, Chonnam National University, Kwangju 500-757, Korea email: sbk@chonnam.chonnam.ac.kr