Some inequalities on totally real submanifolds in locally conformal Kaehler space forms

J. Korean Math. Soc. 41 (2004), No. 5, pp. 795–808 SOME INEQUALITIES ON TOTALLY REAL SUBMANIFOLDS IN LOCALLY CONFORMAL KAEHLER SPACE FORMS Alfonso Carriazo, Young Ho Kim∗and Dae Won Yoon Abstract. In this article, we establish sharp relations between the sectional curvature and the shape operator and also between the k-Ricci curvature and the shape operator for a totally real submanifold in a locally conformal Kaehler space form of constant holomorphic sectional curvature with arbitrary codimension. 1. Introduction Nash’s Theorem enables us to consider any Riemannian manifold as a submanifold of Euclidean space. This gives us a natural motivation for the study of submanifolds of Riemannian manifolds. In this case, we have intrinsic invariants as well as extrinsic invariants. Among extrinsic invariants, the shape operator and the squared mean curvature are the most important ones. Among the main intrinsic invariants, sectional, Ricci and scalar curvature are the well-known ones. Gauss- Bonnet Theorem, isoperimetric inequality and Chern-Lashof Theorem provide relations between intrinsic invariants and extrinsic invariants for a submanifold in a Euclidean space. B.-Y. Chen ([1, 2]) established a inequality relating intrinsic quantities and extrinsic ones for submanifolds in a space form with arbitrary codimension. In particular, in [1] he investigated a relation between the sectional curvature and the shape operator for submanifolds in Riemannian space forms. And, in [2] he established a sharp relation between the k-Ricci curvature and the shape operator. On the other hand, for the above mentioned contents K. Matsumoto, I. Mihai and A. Oiaga ([6]) Received April 10, 2003. 2000 Mathematics Subject Classification: 53B25, 53B35, 53C40. Key words and phrases: mean curvature, sectional curvature, shape operator, k- Ricci curvature, locally conformal Kaehler space form, totally real submanifold. ∗supported by Korea Research Foundation Grant(KRF-2001-015-DP0033). 796 Alfonso Carriazo, Young Ho Kim and Dae Won Yoon studied these relations of slant submanifolds in complex space forms, and Y. H. Kim, D. W. Yoon and C. W. Lee ([4]) have recently investigated these relations of slant submanifolds in Sasakian spaces. In this paper, we study submanifolds of locally conformal Kaehler space forms of constant holomorphic sectional curvature with arbitrary codimension and establish relations between the sectional curvature and the shape operator and also between the k-Ricci curvature and the shape operator for totally real submanifolds in locally conformal Kaehler space forms. 2. Preliminaries Let ˜ Mbe a Hermitian manifold with almost complex structure Jand a Hermitian metric g. A Hermitian manifold ˜ Mis called a locally conformal Kaehler manifold if each point p∈˜ Mhas an open neighborhood Uwith a differentiable map φ:U−→ Rsuch that (2.1) g∗=e−2φg|U is Kaehler metric on U(See [3, 7]). On the other hand, the fundamental 2-form wof ˜ Mis defined by (2.2) w(X, Y ) = g(JX, Y ) for any tangent vectors X, Y on ˜ M. Proposition 2.1 ([3]). A Hermitian manifold ˜ Mis a locally conformal Kaehler manifold if and only if there exists a global closed 1-form αsatisfying (2.3) (˜ ∇Zw)(X, Y ) =β(Y)g(X, Z)−β(X)g(Y, Z) + α(Y)w(X, Z)−α(X)w(Y, Z) for any tangent vectors X, Y, Z on ˜ M, where ˜ ∇denotes the Levi-Civita connection with respect to gand the 1-form βis given by β(X) = −α(JX). The 1-form αwhich satisfies is called the Lee form and its dual vector field is the Ä Lee vector field. A locally conformal Kaehler manifold having the parallel Lee form is called a generalized Hopf manifold. As a matter of fact, the Hopf manifold diffeomorphic to S1×S2n−1is an example of a locally conformal Kaehler manifold that is not Kaehlerian. Some inequalities 797 On a locally conformal Kaehler manifold, a symmetric (0,2)-tensor Pis defined by (2.4) P(X, Y ) = −(˜ ∇Xα)Y−α(X)α(Y) + 1 2||α||2g(X, Y ), and another skew-symmetric (0,2)-tensor ˜ Pby ˜ P(X, Y ) = P(JX, Y ), where ||α|| is the norm of αwith respect to g. Let Mbe an n-dimensional submanifold of an m-dimensional locally conformal Kaehler manifold ˜ M. Let ∇be the induced Levi-Civita connection on M. Then the Gauss and Weingarten formulas are given respectively by (2.5) ˜ ∇XY=∇XY+h(X, Y ), (2.6) ˜ ∇XV=−AVX+DXV for vector fields X, Y tangent to Mand a vector field Vnormal to M, where hdenotes the second fundamental form, Dthe normal connection and AVthe shape operator in the direction of V. The second fundamental form and the shape operator are related by (2.7) g(h(X, Y ), V ) = g(AVX, Y ). We also use gfor the induced Riemannian metric on Mas well as the locally conformal Kaehler manifold ˜ M. Moreover, the mean curvature vector Hon Mis defined by H=1 ntraceh. A submanifold Min ˜ Mis called totally geodesic if the second fundamental form vanishes identically and totally umbilical if there is a real number λsuch that h(X, Y ) = λg(X, Y )Hfor any tangent vectors X, Y on M. For an n-dimensional Riemannian manifold M, we denote by K(π) the sectional curvature of Massociated with a plane section π⊂TpM, p ∈M. For an orthonormal basis {e1, . . . , en}of the tangent space TpM, the scalar curvature τand the normalized scalar curvature ρare defined respectively by τ=X i<j Kij, ρ =2τ n(n−1), where Kij denotes the sectional curvature of the 2-plane section spanned by ei, ej. Suppose Lis a k-plane section of TpMand Xa unit vector in L. We choose an orthonormal basis {e1, . . . , ek}of Lsuch that e1=X. Define the Ricci curvature RicLof Lat Xby (2.8) RicL(X) = K12 +· · · +K1k. 798 Alfonso Carriazo, Young Ho Kim and Dae Won Yoon We simply call such a curvature a k-Ricci curvature. The scalar curvature τof the k-plane section Lis given by (2.9) τ(L) = X 1≤i<j≤k Kij. For each integer k, 2≤k≤n, the Riemannian invariant Θkon an n-dimensional Riemannian manifold Mis defined by (2.10) Θk(p) = 1 k−1inf L,X RicL(X), p ∈M, where Lruns over all k-plane sections in TpMand Xruns over all unit vectors in L. Recall that for a submanifold Min a Riemannian manifold, the relative null space or the kernel of the second fundamental form of Mat a point p∈Mis defined by (2.11) Np={X∈TpM|h(X, Y ) = 0 for all Y∈TpM}. 3. Totally real submanifolds in locally conformal Kaehler space forms Let Mbe an n-dimensional submanifold isometrically immersed in an m-dimensional locally conformal Kaehler manifold ˜ M. A locally conformal Kaehler manifold ˜ Mis said to be a locally conformal Kaehler space form if the holomorphic sectional curvature of the 2-plane section {X, JX}at each point of ˜ Mis a real constant ˜calong ˜ M. A locally conformal Kaehler space form will be denoted by ˜ M(˜c). Then, the Riemannian curvature tensor ˜ Ron ˜ M(˜c) is given by (3.1) ˜ R(X, Y )Z=˜c 4{g(Y, Z)X−g(X, Z)Y+w(Y, Z)JX −w(X, Z)JY −2w(X, Y )JZ} +3 4{g(Y, Z)P1X−g(X, Z)P1Y+P(Y, Z)X−P(X, Z)Y} −1 4{w(Y, Z)˜ P1X−w(X, Z)˜ P1Y+˜ P(Y, Z)JX −˜ P(X, Z)JY −2˜ P(X, Y )JZ −2w(X, Y )˜ P1Z}, where g(P1X, Y ) = P(X, Y ), g(˜ P1X, Y ) = ˜ P(X, Y ). A submanifold Misometrically immersed in ˜ M(˜c) is called totally real if the almost complex structure Jof ˜ M(˜c) carries each tangent space of Some inequalities 799 Minto its corresponding normal space. On the other hand, for a totally real submanifold Mon ˜ M(˜c) we have w(X, Y ) = 0 for vector fields X, Y tangent to M. We denote by Rthe Riemannian curvature tensor field of M. Then, the equation of Gauss on Mis given by (3.2) g(R(X, Y )Z, W) = ˜c 4{g(X, W)g(Y, Z)−g(X, Z)g(Y, W )} +3 4{g(X, W)P(Y, Z)−g(Y, W )P(X, Z) + P(X, W)g(Y, Z) −P(Y, W )g(X, Z)}+g(h(X, W ), h(Y, Z)) −g(h(X, Z), h(Y, W )), for any vector fields X, Y, Z, W tangent to M. Let {e1, . . . , en}be any orthonormal basis in TpM. Then it is easily seen that the scalar curvature τof Mat pis obtained by (3.3) 2τ(p) = n2||H||2− ||h||2+1 4n(n−1)(˜c+ 6σ), where ||H||2and ||h||2are the squared mean curvature and the squared norm of the second fundamental form respectively, and we have put σ=1 nPn i=1 P(ei, ei). 4. Sectional curvature and shape operator B.-Y. Chen ([1]) established a relation between the sectional curvature and the shape operator for submanifolds in real space forms. Also, K. Matsumoto, I. Mihai and A. Oiaga ([6]) and Y. H. Kim, D. W. Yoon and C. W. Lee ([4]) have recently investigated these relations for slant submanifolds into complex space forms and Sasakian spaces, respectively. We prove a similar inequality for an n-dimensional totally real submanifold Minto an m-dimensional locally conformal Kaehler space form ˜ M(˜c) of constant holomorphic sectional curvature ˜c. Lemma 4.1. Let x:M−→ ˜ M(c)be an isometric immersion of an n-dimensional totally real submanifold with normalized scalar curvature ρinto an m-dimensional locally conformal Kaehler space form ˜ M(˜c)of constant holomorphic sectional curvature ˜c. Then, we have (4.1) ||H||2≥ρ−1 4(˜c+ 6σ), equality holding at a point p∈Mif and only if pis a totally umbilical point. 800 Alfonso Carriazo, Young Ho Kim and Dae Won Yoon Proof. Let pbe a point of M. We choose an orthonormal basis {e1, e2, . . . , en}for the tangent space TpMand {en+1, . . . , e2m}for the normal space T⊥ pMat psuch that the normal vector en+1 is in the direction of the mean curvature vector and e1, e2, . . . , endiagonalize the shape operator An+1. Then we have (4.2) An+1 =       a10 0 . . . 0 0a20. . . 0 0 0 a3. . . 0 . . .. . .. . ..... . . 0 0 0 . . . an        , Ar= (hr ij), n X i=1 hr ii = 0,1≤i, j ≤n;n+ 2 ≤r≤2m. From the equation of Gauss (3.2) (4.3) n2||H||2= 2τ+ n X i=1 a2 i+ 2m X r=n+2 n X i,j=1 (hr ij)2−1 4n(n−1)(˜c+ 6σ). On the other hand, (4.4) X i<j (ai−aj)2= (n−1) n X i=1 a2 i−2X i<j aiaj. Therefore, from the above equation we have (4.5) n2||H||2= ( n X i=1 ai)2= n X i=1 a2 i+ 2 X i<j aiaj≤n n X i=1 a2 i. Combining (4.3) and (4.5) (4.6) n(n−1)||H||2≥2τ−1 4n(n−1)(˜c+ 6σ) + 2m X r=n+2 n X i,j=1 (hr ij)2 which implies inequality (4.1). If the equality sign of (4.1) holds at a point p∈Mthen from (4.4) and (4.6) we get Ar= 0 (r=n+2, . . . , 2m) and a1=· · · =an.Consequently, pis a totally umbilical point. The converse is trivial. Theorem 4.2. Let x:M−→ ˜ M(˜c)be an isometric immersion of an n-dimensional totally real submanifold Minto an m-dimensional locally conformal Kaehler space form ˜ M(˜c)of constant holomorphic sectional curvature ˜c. If there exist a point p∈Mand a number c > 1 4(˜c+ 6σ) Some inequalities 801 such that inf K(p) = K≥cat p. Then the shape operator at the mean curvature vector satisfies (4.7) AH>n−1 n½c−1 4(˜c+ 6σ)¾Inat p, where Inis the identity map of TpM. Proof. Assume that Mis a totally real submanifold in ˜ M(˜c). Let p∈Mand a number c > 1 4(˜c+ 6σ) such that K≥cat p. Choose an orthonormal basis {e1, . . . , en, en+1, . . . , e2m}at psuch that en+1 is parallel to the mean curvature vector Hand e1, . . . , endiagonalize the shape operator An+1. Then we have the relation (4.2). We have put uij =uji =aiaj.From the Gauss’s equation we get (4.8) uij ≥c−1 4(˜c+ 6σ) + 2m X r=n+2 (hr ij)2− 2m X r=n+2 hr iihr jj,1≤i6=j≤n. We need the following lemmas in order to complete the proof of the theorem. Lemma 4.3. The following statements hold. (1) For any fixed i∈ {1, . . . , n},we have Pj6=iuij ≥(n−1){c−1 4(˜c+ 6σ)}. (2) uij 6= 0 for i6=j. (3) For distinct i, j, k, we have a2 i=uijuiku−1 jk . Proof. From (4.2) and (4.8), we get n X j6=i uij ≥(n−1){c−1 4(˜c+ 6σ)}+ 2m X r=n+2 {X j6=i (hr ij)2−hr ii X j6=i hr jj} = (n−1){c−1 4(˜c+ 6σ)}+ 2m X r=n+2 X i,j (hr ij)2, which yields statement (1). For statement (2), assume that uij = 0 for i6=j, then ai= 0 or aj= 0. ai= 0 implies that uit = 0 for any i6=t. Hence Pi6=tuit = 0 which contradicts statement (1). Statement (3) follows from uijuik =a2 iajak=a2 iujk. We put Sk={B⊂ {1, . . . , n}:|B|=k}. For any B∈Skwe denote by ¯ B={1, . . . , n}\B. 802 Alfonso Carriazo, Young Ho Kim and Dae Won Yoon Lemma 4.4. For a fixed k,1≤k≤[n 2], and each B∈Sk, we have X j∈BX t∈¯ B ujt ≥(n−k)k{c−1 4(˜c+ 6σ)}. Proof. Without loss of generality, we may assume B={1, . . . , k}. From (4.8) we find X j∈BX t∈¯ B ujt ≥(n−k)k{c−1 4(˜c+ 6σ)}+ 2m X r=n+2 k X j=1 n X t=k+1 {(hr jt)2−hr jjhr tt} = (n−k)k{c−1 4(˜c+ 6σ)}+ 2m X r=n+2 { k X j=1 n X t=k+1 (hr jt)2+ k X j=1 (hr jj)2}, which implies the lemma. Lemma 4.5. For any 1≤i6=j≤n, we have uij >0. Proof. Assume u1n<0. Then, by statement (3) of Lemma 4.3, we get u1iuin <0 for 1 < i < n. Without loss of generality, we may assume (4.9) (u12, . . . , u1l, u(l+1)n, . . . , u(n−1)n>0, u1(l+1), . . . , u1n, u2n, . . . , uln <0, for some [n+1 2]≤l≤n−1. Let l=n−1, then u1n+u2n+· · · +u(n−1)n<0 which contradicts to statement (1) of Lemma 4.3. Thus, l < n −1. From statement (3) of Lemma 4.3, we get (4.10) a2 n=uinutn uit >0, where 2 ≤i≤land l+ 1 ≤t≤n−1. By (4.9) and (4.10) we have uit <0 which implies l X i=1 n X t=l+1 uit = l X i=2 n−1 X t=l+1 uit + l X i=1 uin + n X t=l+1 u1t<0. This contradicts Lemma 4.4. Now, we return to the proof of Theorem 4.2. From Lemma 4.5, it follows that a1, . . . , anare of the same sign. Therefore, the shape Some inequalities 803 operator AHis positive-definite. Assume aj>0 for all j∈ {1, . . . , n}. Then from statement (1) of Lemma 4.3, we obtain nai||H|| − a2 i=ai(a1+· · · +an)−a2 i =aiX i6=j aj=X i6=j aiaj=X i6=j uij ≥(n−1){c−1 4(˜c+ 6σ)}, which implies (4.7). This completes the proof of the theorem. 5. k-Ricci curvature and shape operator In this section, we establish a relation between the k-Ricci curvature and the shape operator for an n-dimensional totally real submanifold Minto an m-dimensional locally conformal Kaehler space form ˜ M(˜c) of constant holomorphic sectional curvature ˜c. Theorem 5.1. Let x:M−→ ˜ M(˜c)be an isometric immersion of an n-dimensional totally real submanifold Minto an m-dimensional locally conformal Kaehler space form ˜ M(˜c)of constant holomorphic sectional curvature ˜c. Then, for any integer k2≤k≤n, and any point p∈M, we have (1) If Θk(p)6=1 4(˜c+ 6σ), then shape operator at the mean curvature satisfies (5.1) AH>n−1 n½Θk(p)−1 4(˜c+ 6σ)¾Inat p, where Indenotes the identity map of TpM. (2) If Θk(p) = 1 4(˜c+ 6σ),then AH≥0at p. (3) A unit vector X∈TpMsatisfies (5.2) AHX=n−1 n½Θk(p)−1 4(˜c+ 6σ)¾X if and only if Θk(p) = 1 4(˜c+ 6σ)and X∈Np. (4) AH=n−1 n{Θk(p)−1 4(˜c+ 6σ)}Inat pif and only if pis a totally geodesic point. Proof. Let {e1, . . . , en}be an orthonormal basis of TpM. Denote by Li1···ikthe k-plane section spanned by ei1, . . . , eik. It follows from (2.8)