Afr. Mat.DOI 10.1007/s13370-014-0223-5

Weyl semisymmetric submanifolds satisfyingChen’s equality

Pradip Majhi

Received: 24 September 2013 / Accepted: 2 January 2014© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014

Abstract The object of the present paper is to study Weyl semisymmetric submanifoldssatisfying Chen’s equality in a Euclidean space.

Keywords Chen invariant · Chen’s inequality · Weyl conformal curvature tensor ·Weyl semisymmetric manifold · Minimal submanifold

Mathematics Subject Classification (2010) 53C40 · 53C25 · 53C42

1 Introduction

One of the basic problems in submanifold theory is to find simple relationships between theextrinsic and intrinsic invariants of a submanifold. In [1] and [4], Chen established inequal-ities in this respect, called Chen inequalities. The main extrinsic invariant is the squaredmean curvature and the main intrinsic invariants include the classical curvature invariants,namely the scalar curvature and the Ricci curvature; and the well known modern curvatureinvariant, namely Chen invariant [2]. In 1993, Chen obtained an interesting basic inequalityfor submanifolds in a real space form involving the squared mean curvature and the Cheninvariant and found several of its applications. This inequality is now well known as Chen’sinequality; and in the equality case it is known as Chen’s equality.

In [6], Dillen, Petrovic and Verstraelen studied Einstein, conformally flat and semisymmet-ric submanifolds satisfying Chen’s equality in Euclidean spaces. The hypersurfaces in E

n+1

satisfying Chen’s equality have been studied in [7] and others. Recently Özgür and De [8]studied projectively semisymmetric submanifolds satisfying Chen’s equality in a Euclideanspace and the submanifold satisfying the condition P.P = 0, where P is the projective cur-

P. Majhi (B)Department of Mathematics, University of North Bengal,Raja Rammohunpur, Darjeeling 734013, West Bengal, Indiae-mail: mpradipmajhi@gmail.com

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P. Majhi

vature tensor. Motivated by the above studies in this paper, we study submanifolds satisfyingChen’s equality and the conditions R.C = 0 in a Euclidean space.

The paper is organized as follows.In Sect. 2, we give some idea about Riemannian submanifolds and Chen’s inequality. In

Sect. 3, we give some priliminaries about Chen’s equality which will be used in the nextSection. Finally Sect. 4 deals with the study of Weyl semisymmetric submanifolds satisfyingChen’s equality in a Euclidean space. As a consequence of the above result we obtain someimportant Theorems.

2 Chen’s ineqality

Let M be an n-dimensional submanifold of an (n + m)-dimensional Euclidean space En+m .

The Gauss and Weingarten formulas are given respectively by

∇̃X Y = ∇X Y + σ(X, Y ) and ∇X N = −AN X + ∇⊥X N

for all X , Y ∈ T (M) and N ∈ T ⊥M , where ∇̃, ∇ and ∇⊥ are respectively the Riemannian,induced Riemannian and induced normal connections in M̃ , M and the normal bundle T Mof M respectively, and σ is the second fundamental form related to the shape operator A by〈σ(X, Y ), N 〉 = 〈AN X, Y 〉. The equation of Gauss is given by

R(X, Y, Z , W ) = 〈σ(X, W ), σ (Y, Z)〉 − 〈σ(X, Z), σ (Y, W )〉 (2.1)

for all X , Y , Z , W ∈ T M , where R is the curvature tensor of M .The mean curvature vector H is given by H = 1

n trace(σ ). The submanifold M is totallygeodesic in Em+n if σ = 0, and minimal if H = 0 [3]. Let {e1, e2, ..., en} be an orthonormaltangent frame field on M . For the plane section ei ∧ e j of the tangent bundle T M spanned bythe vectors ei and e j (ei �= e j ) the scalar curvature of M is defined by κ = ∑n

i, j=1 K (ei ∧e j ),where K denotes the sectional curvature of M . Consider the real function inf K on Mn definedfor every x ∈ M by

(inf K )(x) := inf{K (π)| π is plane in Tx Mn}.Note that since the set of planes at a certain point is compact, this infimum is actuallya minimum.

Lemma 2.1 [1] Let M, n ≥ 2, be any submanifold of Em+n. Then

inf K ≥ 1

2

{

κ − n2(n − 2)

n − 1|H |2

}

. (2.2)

Equality holds in (2.2) at a point x if and only if with respect to suitable local orthonormalframes e1,e2,...,en ∈ Tx Mn, the Weingarten maps At with respect to the normal sectionsξt = en+t , t = 1, 2, ..., p are given by

A1 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

a 0 0 0 · · · 00 b 0 0 · · · 00 0 μ 0 · · · 00 0 0 μ · · · 0...

......

.... . .

...

0 0 0 0 · · · μ

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

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Weyl semisymmetric submanifolds satisfying Chen’s equality

At =

⎡

⎢⎢⎢⎢⎢⎣

ct dt 0 · · · 0dt −ct 0 · · · 00 0 μ · · · 0...

......

.... . .

0 0 0 · · · 0

⎤

⎥⎥⎥⎥⎥⎦

(t > 1), where μ = a + b for any such frame, inf K (x) is attained by the plane e1 ∧ e2.The inequality (2.2) is well known as Chen’s inequality. In case of equality, it is known as

Chen’s equality. For dimension n = 2, the Chen’s equality is always true.

3 Preliminaries

Let M be an n-dimensional (n ≥ 3) submanifold of a Euclidean space En+m satisfyingChen’s equality. Then, from Lemma 2.1 we immediately have the following

K12 = ab −m∑

r=1

(c2r + d2

r ), (3.1)

K1 j = aμ, (3.2)

K2 j = bμ, (3.3)

Ki j = μ2, (3.4)

S(e1, e1) = K12 + (n − 2)aμ, (3.5)

S(e2, e2) = K12 + (n − 2)bμ, (3.6)

S(ei , ei ) = (n − 2)μ2, (3.7)

r = 2K12 + (n − 1)(n − 2)μ2, (3.8)

where i , j > 2. Furthermore, R(ei , e j )ek = 0 if i , j and k are mutually different [6].Let (M, g) be an n-dimensional (n ≥ 3) Riemannian manifold. The conformal curvature

tensor C of type (1,3) is defined by [9]

C(X, Y )Z = R(X, Y )Z − 1

n − 2[S(Y, Z)X − S(X, Z)Y (3.9)

+g(Y, Z)Q X − g(X, Z)QY ]+ r

(n − 1)(n − 2)[g(Y, Z)X − g(X, Z)Y ],

where S is the Ricci tensor, Q is the Ricci operator defined by S(X, Y ) = g(Q X, Y ) and ris the scalar curvature of the manifold M . The tensor field C vanishes identically for n = 3.The Weyl conformal curvature tensor C is invariant under any conformal change of metric.A Riemannian manifold is said to be conformally flat if and only if C = 0 for n > 3 andC = 0 for n = 3. Also a Riemannian manifold (n > 3) is said to be Weyl semisymmetric ifits conformal curvature tensor C satisfies the relation R.C = 0.

Using (3.1), (3.2), (3.3), (3.4), (3.5), (3.6), (3.7), (3.8) in (3.9), we have the following:

Proposition 3.1 Let M be an n-dimensional (n > 3) submanifold in a Euclidean spacesatisfying Chen’s equality, then

C122 = (n − 3)

(n − 1)K12e1, (3.10)

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C133 = − (n − 3)

(n − 1)(n − 2)K12e1, (3.11)

C131 = (n − 3)

(n − 1)(n − 2)K12e3, (3.12)

C233 = − (n − 3)

(n − 1)(n − 2)K12e2, (3.13)

C211 = (n − 3)

(n − 1)K12e2, (3.14)

C232 = (n − 3)

(n − 1)(n − 2)K12e3, (3.15)

and

Ci jk = 0 (3.16)

if i , j , k are mutually different.

4 Weyl semisymmetric submanifold of A Euclidean space En+m satisfying Chen’s

equality

In this section we study Weyl semisymmetric submanifold M of a Euclidean space En+m

satisfying Chen’s equality. Therefore R.C = 0 holds on M . Thus we can write

(R(e1, e3).C)(e2, e3)e1 = R(e1, e3)C(e2, e3)e1 − C(R(e1, e3)e2, e3)e1

−C(e2, R(e1, e3)e3)e1 − C(e2, e3)R(e1, e3)e1 = 0 (4.1)

and

(R(e2, e3).C)(e1, e3)e2 = R(e2, e3)C(e1, e3)e2 − C(R(e2, e3)e1, e3)e2

−C(e1, R(e2, e3)e3)e2 − C(e1, e3)R(e2, e3)e2 = 0. (4.2)

Using (3.1), (3.2), (3.3), (3.4), (3.5), (3.6), (3.7), (3.8) and (3.10), (3.11), (3.12), (3.13), (3.14),(3.15), (3.16) we have

aμK12n − 3

n − 2= 0 (4.3)

and

bμK12n − 3

n − 2= 0. (4.4)

Case 1 If μ = 0, then M is minimal.

Case 2 If μ �= 0 and a = 0.

Then μ = a + b = b �= 0. Therefore from (4.4) we have

bK12 = 0.

This implies b∑m

r=1(c2r + d2

r ) = 0, hence cr = dr = 0. So, by [5], M is a round hyperconein some totally geodesic subspace E

n+1 of En+m .

Case 3 If μ �= 0 and b = 0Similar as Case 2.

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Weyl semisymmetric submanifolds satisfying Chen’s equality

Case 4 If μ �= 0, a �= 0, b �= 0In this case if we subtract (4.4) from (4.3), then μ(a − b)K12 = 0. This implies either

a = b or, K12 = 0, since μ �= 0.

Sub Case 1 If K12 = 0, then inf K = 0 [6].

Sub Case 2 If a = b, then μ = a + b = 2a. Thus from (4.3), we have aμ = 0 i.e., 2a2 = 0.Therefore a = 0 which is a contradiction from our assumption.

Case 5 If a = b = 0, then M is a submanifold in some totally geodesic subspace En+m−1,

which has shape operators of the form of the Lemma (2.1).

Therefore in view of the above cases we have the following:

Theorem 4.1 Let M be an n-dimensional (n > 3) submanifold of a Euclidean space En+m

satisfying Chen’s equality. If M is Weyl semisymmetric then

(i) M is minimal, or(ii) inf K = 0, or

(iii) M is a round hypercone in some totally geodesic subspace En+1 of E

n+m, or(iv) M is a submanifold in some totally geodesic subspace E

n+m−1, which has shape oper-ators of the form of the Lemma (2.1).

Suppose n > 3. If M is a round hypercone in some totally geodesic subspace En+1

of En+m, then cr = dr = 0. This implies

K12 = ab −m∑

r=1

(c2r + d2

r ) = ab.

Now subtracting (4.4) from (4.3), we have μK12(a − b) = 0. Therefore either K12 = 0 or,a = b.

Now for K12 = 0 implies inf K = 0, hence M is conformally flat [6]. Therefore R.C = 0.If a = b, then we have K12 = ab = a2 �= 0 and μ = 2a = 2b(�= 0). Therefore from

(4.3) or, (4.4) we have n = 3, which is a contradiction. Also if M is minimal, then R.C = 0.In view of the above result we can state the following:

Theorem 4.2 A n-dimensional (n > 3) submanifold M of a Euclidean space En+m satisfying

Chen’s equality is Weyl semisymmetric if and only if it is either minimal or, round hyperconein some totally geodesic subspace E

n+1 of En+m.

Again in [6] Dillen, Petrovic and Verstraelen proved that a n-dimensional (n ≥ 3) sub-manifold M of a Euclidean space E

n+m satisfying Chen’s equality is semisymmetric if andonly if it is either minimal or, a round (in which case M is (n − 2)-ruled) hypercone in sometotally geodesic subspace E

n+1 of En+m .

Therefore in view of the above cases we have the following:

Theorem 4.3 For an n-dimensional (n > 3) submanifold M of a Euclidean space En+m

satisfying Chen’s equality the following are equivalent

(i) R.R = 0 on M(ii) R.C = 0 on M

(iii) either M is minimal or, a round hypercone in some totally geodesic subspace En+1

of En+m .

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P. Majhi

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