Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds

Transcription

1 INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 0 NO. 2 PAGE ) Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds Andreea Olteanu Communicated by Ion Miai) ABSTRACT Recently, te autor establised a general inequality for doubly warped products in arbitrary Riemannian manifolds [6]. In te present paper, we obtain similar inequalities for doubly warped products isometrically immersed in locally conformal almost cosymplectic manifolds. Some applications are derived. Keywords: Doubly warped product; minimal immersion; totally real submanifold; locally conformal almost cosymplectic manifold. AMS Subject Classification 200): Primary: 53C0 ; Secondary: 53C25; 53B25; 53C2.. Introduction Bisop and O Neill [] introduced te concept of warped products to study manifolds of negative sectional curvature. O Neill discussed warped products and explored curvature formulas of warped products in terms of curvatures of components of warped products. Doubly warped products can be considered as a generalization of singly warped products. Let M, g ) and M 2, g 2 ) be two Riemannian manifolds and let f : M 0, ) and f 2 : M 2 0, ) be differentiable functions. Te doubly warped product M = f2 M f M 2 is te product manifold M M 2 endowed wit te metric g = f 2 2 g + f 2 g 2. More precisely, if π : M M 2 M and π 2 : M M 2 M 2 are natural projections, te metric g is defined by g = f 2 π 2 ) 2 π g + f π ) 2 π 2g 2. Te functions f and f 2 are called warping functions. If eiter f or f 2, but not bot, ten we obtain a warped product. If bot f and f 2, ten we ave a Riemannian product manifold. If neiter f nor f 2 is constant, ten we ave a non-trivial doubly warped product [20]. Let x : f2 M f M 2 M be an isometric immersion of a doubly warped product f2 M f M 2 into a Riemannian manifold M. We denote by te second fundamental form of x and by Hi = n i trace i te partial mean curvatures, were trace i is te trace of restricted to M i and n i = dim M i i =, 2). Te immersion x is said to be mixed totally geodesic if X, Z) = 0, for any vector fields X and Z tangent to D and D 2, respectively, were D i are te distributions obtained from te vectors tangent to M i or more precisely, vectors tangent to te orizontal lifts of M i ). In [], B. Y. Cen studied te relationsip between te Laplacian of te warping function f and te squared mean curvature of a warped product M f M 2 isometrically immersed in a Riemannian manifold M c) of constant sectional curvature c given by f f n2 n 2 H 2 + n c,.) were n i = dim M i, i =, 2, and is te Laplacian operator of M. Moreover, te equality case of.) olds if and only if x is a mixed totally geodesic immersion and n H = n 2 H 2, were H i, i =, 2, are te partial mean curvature vectors. Received : , Accepted : * Corresponding autor

2 Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds Later, several Cen inequalities ave been obtained for different submanifolds in ambient manifolds possessing different kind of structures see [7], [8], [0], [], [3], []). In 2008, B. Y. Cen and F. Dillen extended inequality.) to multiply warped product manifolds in arbitrary Riemannian manifolds see [5]). In [6], te present autor establised te following general inequality for arbitrary isometric immersions of doubly warped product manifolds in arbitrary Riemannian manifolds: Teorem.. Let x be an isometric immersion of an n-dimensional doubly warped product M = f2 M f M 2 into an m-dimensional arbitrary Riemannian manifold M m. Ten: f f 2 H 2 + n n 2 max K,.2) were n i = dim M i, n = n + n 2, i is te Laplacian operator of M i, i =, 2 and max K p) denotes te maximum of te sectional curvature function of M m restricted to 2-plane sections of te tangent space T p M of M at eac point p in M. Moreover, te equality case of.2) olds if and only if te following two statements old:. x is a mixed totally geodesic immersion satisfying n H = n 2 H 2, were H i, i =, 2, are te partial mean curvature vectors of M i. 2. at eac point p = p, p 2 ) M, te sectional curvature function K of M m satisfies K u, v) = max K p) for eac unit vector u T p M and eac unit vector v T p2 M 2. Later, in [8] and [9], te present autor studied doubly warped product submanifolds in generalized Sasakian space forms and S-space forms, respectively. In [6], M. Crasmareanu introduced and studied biwarped 3-metrics from te point of view of Webster curvature. In [2], D. W.Yoon, K. S. Co, S. G. Han investigated warped product submanifolds in locally conformal almost cosymplectic manifolds. Motivated by te studies of te above autors, te aim of tis paper is to obtain similar inequalities for doubly warped products in locally conformal almost cosymplectic manifolds of pointwise constant ϕ-sectional curvature c. 2. Preliminaries In tis section, we recall some general definitions and basic formulas wic will be used later. For more background on locally conformal almost cosymplectic manifolds, we recommend te reference [5]. Let M be a 2m + )-dimensional almost contact manifold see [2]). Denote by ϕ, ξ, η) te almost contact structure of M. Tus, ϕ is a, )tensor field, ξ is a vector field and η a -form on M suc tat ϕ 2 = I + η ξ, ηξ) =. Define an almost complex structure J on te product manifold M R by JX, λ d ds ) = ϕx λξ, ηx) d ), 2.) ds were X and λ d ds are vectors tangent to M and R, respectively, R being te real line wit coordinate s. Te manifold M is said to be normal [2]) if te almost complex structure J is integrable i.e., J arises from a complex structure on M R). Te necessary and sufficient condition for M to be normal is were [ϕ, ϕ] is te Nijenuis tensor of ϕ defined by [ϕ, ϕ] + 2dη ξ = 0, [ϕ, ϕ]x, Y ) = [ϕx, ϕy ] ϕ[ϕx, Y ] ϕ[x, ϕy ] + ϕ 2 [X, Y ] for any vector fields X and Y tangent to M, [, ] being te Lie bracket of vector fields. Let g be a Riemannian metric on M compatible wit ϕ, ξ, η), suc tat gϕx, ϕy ) = gx, Y ) ηx)ηy ) 7

3 A. Olteanu for any vector fields X and Y tangent to M. Tus, te manifold M is almost contact metric and te quadruple ϕ, ξ, η, g) is its almost contact metric structure. Clearly, we ave ηx) = gx, ξ) for any vector field X tangent to M. Let Φ denote te fundamental 2-form of M defined by ΦX, Y ) = gϕx, Y ) for any vector fields X and Y tangent to M. Te manifold M is said to be almost cosymplectic if te forms η and Φ are closed, i.e., dη = 0 and dφ = 0, were d is te operator of exterior differentiation. If M is almost cosymplectic and normal, ten it is called cosymplectic [2]). It is well known tat te almost contact metric manifold is cosymplectic if and only if ϕ vanises identically, were is te Levi-Civita connection on M. Te products of almost Käler manifolds and R are te simplest examples of almost cosymplectic manifolds. An almost contact metric manifold M is called a locally conformal almost cosymplectic manifold [5]) if tere exists a -form ω suc tat dφ = 2ω Φ, dη = ω η and dω = 0. A necessary and sufficient condition for a structure to be normal locally conformal almost cosymplectic is [9]) X ϕ)y = ugϕx, Y )ξ ηy )ϕx), 2.2) were ω = uη. From 2.2) it follows tat X ξ = ux ηx)ξ). A plane section σ in T p M of an almost contact structure manifold M is called a ϕ-section if σ ξ and ϕσ) = σ. M is of pointwise constant ϕ- sectional curvature if at eac point p M, te section curvature Kσ) does not depend on te coice of te ϕ- section σ of T p M and, in tis case for p M and for any ϕ- section σ of Tp M, te function c defined by cp) = Kσ) is called te ϕ- sectional curvature of M. A locally conformal almost cosymplectic manifold M of dimension 5 is of pointwise constant ϕ- sectional curvature if and only if its curvature tensor R is of te form [5]) RX, Y, Z, W ) = c 3u2 {gx, W )gy, Z) gx, Z)gY, W )} 2.3) + c + u2 {gx, ϕw )gy, ϕz) gx, ϕz)gy, ϕw ) 2gX, ϕy )gz, ϕw )} c + u2 + u ){gx, W )ηy )ηz) gx, Z)ηY )ηw )} +gy, Z)ηX)ηW ) gy, W )ηx)ηz)} +gx, W ), Y, Z)) gx, Z), Y, W )), were u is a function suc tat ω = uη, u = ξu and c is te pointwise constant ϕ- sectional curvature of M. Let M be an n-dimensional manifold isometrically immersed in a locally conformal almost cosymplectic manifold M. Let be te induced Levi-Civita connection of M. Ten te Gauss and Weingarten formulas are given respectively by X Y = X Y + X, Y ), X V = A V X + D X V for vector fields X, Y tangent to M and a vector field V normal to M, were denotes te second fundamental form, D te normal connection and A V te sape operator in te direction of V. Te second fundamental form and te sape operator are related as gx, Y ), V ) = ga V X, Y ). We also denote by g te induced Riemannian metric on M as well as on te locally conformal almost cosymplectic manifold M. Let X be tangent to M we put ϕx = P X + F X, were P X and F X are te tangential and te normal components of ϕx, respectively. Te squared norm of P for an otonormal basis {e,..., e n } of M is defined by P 2 = n g 2 P e i, e j ) i,j= 75

4 Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds and te mean curvature vector Hp) at p M is given by H = n trace = n As is known, M is said to be minimal if H vanises identically. Also, for any r {n +,..., 2m + } we set n e i, e i ). r ij = g e i, e j ), e r ), i, j {,..., n}, and n 2 = g e i, e j ), e i, e j )) i,j= were {e n+,..., e } is an ortonormal basis of T p M. A submanifold M is totally geodesic in M if = 0 and totally real submanifold if P is identically zero, tat is, ϕx Tp M for any X T p M, p M. For an n-dimensional Riemannian manifold M, we denote by Kπ) te sectional curvature of M associated wit a plane section π T p M, p M. For any ortonormal basis e,..., e n of te tangent space T p M, te scalar curvature τ at p is defined by τp) = K e i e j ). i<j n Let M be a Riemannian p-manifold and {e,..., e p } be an ortonormal basis of M. For a differentiable function f on M, te Laplacian f of f is defined by f = p { ) ej e j f ej e j f}. We recall te following general algebraic lemma of Cen for later use. Lemma 2.. [3]Let n 2 and a, a 2,..., a n, b real numbers suc tat j= n ) 2 n ) a i = n ) a 2 i + b. Ten 2a a 2 b, wit equality olding if and only if a + a 2 = a 3 =... = a n. 3. Doubly warped product submanifolds in locally conformal almost cosymplectic manifolds Next, we investigate doubly warped product submanifolds tangent to te structure vector field ξ in a locally conformal almost cosymplectic manifold M c). Teorem 3.. Let x be an isometric immersion of an n-dimensional doubly warped product f2 M f M 2 into a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M. Ten: f f 2 H 2 + n n 2 c + u2 were n i = dim M i and i is te Laplacian operator of M i, i =, 2. + u )n 2 + 3c + u2 ) n 2, 3.) 76

5 A. Olteanu Proof. Suppose tat f2 M f M 2 is a doubly warped product submanifold of a locally conformal almost cosymplectic manifold M c) wit pointwise constant ϕ- sectional curvature c wose structure vector field ξ is tangent to M. Since f2 M f M 2 is a doubly warped product, ten { X Y = X Y f 2 2 g f 2 X, Y ) 2 ln f 2 ), 3.2) X Z = Z ln f 2 ) X + X ln f ) Z, for any vector fields X, Z tangent to M and M 2, respectively, were and 2 are te Levi-Civita connections of te Riemannian metrics g and g 2, respectively. Here, 2 ln f 2 ) denotes te gradient of ln f 2 wit respect to te metric g 2. If X and Z are unit vector fields, it follows tat te sectional curvature K X Z) of te plane section spanned by X and Z is given by K X Z) = f { XX ) f X 2 f } + f 2 { 2 ZZ ) f 2 Z 2 f 2 }. 3.3) We coose a local ortonormal frame {e,..., e n, e n+,..., e } suc tat e,..., e n = ξ are tangent to M, e n+,..., e n are tangent to M 2 and e n+ is parallel to te mean curvature vectore H. Ten from 3.3), it follows tat Using te equation of Gauss, we get j n n + s n n 2 H 2 = 2τ + 2 n n ) c 3u2 were τ denotes te scalar curvature of f2 M f M 2, tat is, τ = K e i e j ). Let us consider tat δ = 2τ n n ) c 3u2 Ten, 3.5) can be written as K e j e s ) = n 2 f f + n 2 f 2 f 2. 3.) i<j n 3c + u2 ) P 2 + c + u2 + u )2n 2), 3.5) 3c + u2 ) P 2 + c + u2 + u )2n 2) n2 2 H ) n 2 H 2 = 2 δ + 2). 3.7) Tus wit respect to te above ortonormal frame, te above equation takes te following form: n ) 2 n ) ii = 2 δ + n+ 2 ) ii + n+ 2 n ) ij + r 2 ij. Assume tat a =, a 2 = n i=2 n+ ii and a 3 = n 3 ) 2 3 a i = 2[δ + a 2 i + i j t=n + n+ tt i j n jj kk 2 j k n Tus a, a 2, a 3 satisfy te Lemma of Cen for n = 3), i.e., 3 ) 2 a i = 2 b + r=n+2 i,j=. Ten te above equation becomes ) n+ 2 ij + n + s t n 3 a 2 i r=n+2 i,j= ss tt ]. ), n ) r 2 ij 77

6 Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds wit b = δ + i j n jj kk 2 j k n ) n+ 2 ij + r=n+2 i,j= n + s t n n ) r 2 ij ss tt. Ten 2a a 2 b, wit equality olding if and only if a + a 2 = a 3. In te case under consideration, it follows tat jj kk + ss tt 3.8) j<k n n + s<t n Equality olds if and only if δ 2 + α<β n n Using again te Gauss equation, we find tat αβ n 2 f f + n 2 f 2 f 2 = τ ii = = τ n n ) ) 8 + c + u2 + u )n ) n 2 n 2 ) ) 8 ) n t=n + n r=n+2 α,β= j<k n K e j e k ) r αβ ) 2. tt. 3.9) n + s<t n K e s e t ) = g 2 P e j, e k ) 3c + u2 ) + j<k n r=n+ j<k n r=n+ n + s<t n n + s<t n r jj r kk ) ) r 2 jk g 2 P e s, e t ) 3c + u2 ) 3.0) r ss r tt r st) 2). Combining 3.8) and 3.0) and taking account of 3.), we obtain n n ) ) c 3u 2 n 2 + n τ + n n 2 3.) f f 2 8 δ + u2 + c + u )n ) 2 g 2 P e j, e k ) 3c + u2 ) g 2 P e s, e t ) 3c + u2 ). j<k n n + s<t n Hence, from 3.6), te inequality 3.) reduces to f f 2 H 2 + n n 2 c + u2 + u )n c + u2 ) n + j<t n g 2 P e j, e t ) n2 H 2 + n n 2 c + u2 + u )n

7 A. Olteanu + 3c + u2 ) min{n, n 2 }. 3.2) We distinguis two cases: a) n n 2, in tis case te inequality 3.2) implies 3.). b) n > n 2, in tis case te inequality 3.2) also becomes 3.). It completes te proof. Remark 3.. If f 2, ten te inequality 3.) is exactly te inequality 6) from [2] for warped products. Corollary 3.. Let x be an isometric immersion of an n-dimensional totally real doubly warped product f2 M f M 2 into a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M. Ten, we ave: f f 2 H 2 + n n 2 c + u2 + u )n 2, 3.3) were n i = dim M i and i is te Laplacian operator of M i, i =, 2. Moreover, te equality case of 3.3) olds if and only if x is a mixed totally geodesic immersion and n H = n 2 H 2, were H i, i =, 2, are te partial mean curvature vectors. Proof. Let f2 M f M 2 be a doubly warped product submanifold of a locally conformal almost cosymplectic manifold M c). Ten we ave gp ei, e s ) = 0 for 0 i n, n + s n. Terefore, by 3.2) we can easily obtain te inequality 3.3). Also, we see tat te equality of 3.2) olds if and only if r jt = 0, j n, n + t n, n + r 2m +, 3.) and n r ii = t=n + n r tt = 0, n + 2 r 2m ) Obviously 3.) is equivalent to te mixed totally geodesic of te doubly warped product f2 M f M 2 and 3.9) and 3.5) imply n H = n 2 H 2. Te converse statement is straigtforward. As applications, we obtain certain obstructions to te existence of minimal totally real doubly warped product submanifolds in locally conformal almost cosymplectic manifolds. By using te above corollary, we can obtain some important consequences: Corollary 3.2. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M. If te warping functions are armonic, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c < n u2 + 3n u 2 + u ). Proof. Assume f is a armonic function on M, f 2 is a armonic function on M 2 and f2 M f M 2 admits a minimal totally real immersion in a locally conformal almost cosymplectic manifold Mc). Ten, te inequality 3.3) becomes c n u2 + 3n u 2 + u ). Corollary 3.3. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M. If te warping functions f and f 2 of f2 M f M 2 are eigenfunctions of te Laplacian on M and M 2, respectively, wit corresponding eigenvalues λ > 0 and λ 2 > 0, respectively, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c n u2 + 3n u 2 + u ). Corollary 3.. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M. If one of te warping functions is armonic and te oter one is an eigenfunction of te Laplacian wit corresponding eigenvalue λ > 0, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c n u2 + 3n u 2 + u ). 79

8 Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds Teorem 3.2. Let x be an isometric immersion of an n-dimensional doubly warped product f2 M f M 2 into a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M 2. Ten: f f 2 H 2 + n n 2 c + u2 were n i = dim M i and i is te Laplacian operator of M i, i =, 2. + u )n + 3c + u2 ) n 2, 3.6) Remark 3.2. If f 2, ten te inequality 3.6) is exactly te inequality from Teorem 8 from [2] for warped products. Corollary 3.5. Let x be an isometric immersion of an n-dimensional totally real doubly warped product f2 M f M 2 into a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M 2. Ten, we ave: f f 2 H 2 + n n 2 c + u2 + u )n, 3.7) were n i = dim M i and i is te Laplacian operator of M i, i =, 2. Moreover, te equality case of 3.7) olds if and only if x is a mixed totally geodesic immersion and n H = n 2 H 2, were H i, i =, 2, are te partial mean curvature vectors. Corollary 3.6. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M 2. If te warping functions are armonic, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c < n 2 u2 + 3n 2 u 2 + u ). Corollary 3.7. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M 2. If te warping functions f and f 2 of f2 M f M 2 are eigenfunctions of te Laplacian on M and M 2, respectively, wit corresponding eigenvalues λ > 0 and λ 2 > 0, respectively, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c n 2 u2 + 3n 2 u 2 + u ). Corollary 3.8. Let f2 M f M 2 be a totally real doubly warped product in a 2m + )-dimensional locally conformal almost cosymplectic manifold of pointwise constant ϕ-sectional curvature c wose structure vector field ξ is tangent to M 2. If one of te warping functions is armonic and te oter one is an eigenfunction of te Laplacian wit corresponding eigenvalue λ > 0, ten f2 M f M 2 admits no minimal totally real immersion into a locally conformal almost cosymplectic manifold M c) wit c n 2 u2 + 3n 2 u 2 + u ). References [] Bisop, R. L. and O Neill, B., Manifolds of negative curvature. Trans. Amer. Mat. Soc ), -9. [2] Blair, D. E., Contact Manifolds in Riemannian Geometry. Lecture Notes in Mat. 509, Springer, Berlin, 976. [3] Cen, B. Y., Some pincing and classification teorems for minimal submanifolds. Arc. Mat ), [] Cen, B. Y., On isometric minimal immersions from warped products into real space forms. Proc. Edinburg Mat. Soc ), [5] Cen, B. Y. and Dillen, F., Optimal inequalities for multiply warped product submanifolds. Int. Electron. J. Geom., Vol. 2008), -. [6] Crasmareanu, M., Adapted metrics and Webster curvature on tree classes of 3-dimensional geometries. Int. Electron. J. Geom., 7 2) 20), [7] Malek, F. and Nejadakbary, V., Warped product submanifold in generalized Sasakian space form. Acta Mat. Acad. Paedagog. Nyazi. N.S.) 27 no. 2 20), [8] Matsumoto, K. and Miai, I., Warped product submanifolds in Sasakian space forms. SUT Journal of Matematics ), 35-. [9] Matsumoto, K., Miai, I. and Rosca, R., A certain locally conformal almost cosymplectic manifold and its submanifolds. Tensor N.S. 5 ) 992), [0] Miai, A., Warped product submanifolds in complex space forms. Acta Sci. Mat. Szeged) ), [] Miai, A., Warped product submanifolds in quaternion space forms. Rev. Roumaine Mat. Pures Appl ), [2] Miai, A., Miai I. and Miron, R. Eds.), Topics in Differential Geometry, Ed. Academiei Romane, Bucuresti, [3] Miai, I. and Presura, I., An improved Cen first inequality for Legendrian submanifolds in Sasakian space forms. Period. Mat. Hung. 7 2) 207), [] Muratan, C., Arslan, K., Ezentas, R. and Miai, I., Warped product submanifolds in Kenmotsu space forms. Taiwanese J. Mat ),

9 A. Olteanu [5] Olszak, Z., Locally conformal almost cosymplectic manifolds. Collq. Mat. 57) 989), [6] Olteanu, A., A general inequality for doubly warped product submanifolds. Mat. J. Okayama Univ ), [7] Olteanu, A., Recent results in te geometry of warped product submanifolds, Matrix Rom, 20. [8] Olteanu, A., Doubly warped product submanifolds in generalized Sasakian space forms, Proceedings RIGA 20, Ed. Univ. Bucuresti 20), 7-8. [9] Olteanu, A., Doubly warped products in S-space forms. Rom. J. Mat. Comput. Sci. Issue 20), -2. [20] Ünal, B., Doubly warped products. Differ. Geom. App. 53) 200), [2] Yoon, D. W., Co, K. S. and Han, S. G., Some inequalities for warped products in locally conformal almost cosymplectic manifolds. Note Mat. 23 ) 200), Affiliations ANDREEA OLTEANU ADDRESS: University of Agronomic Sciences and Veterinary Medicine of Bucarest, Dept. of Matematics, Pysics and Terrestrial Measurements, 06, Bucarest-Romania. andreea_d_olteanu@yaoo.com ORCID ID : orcid.org/