Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form

28/10/2015
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OAI : oai:www.see.asso.fr:11784:14282

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In this article, we derive an inequality satisfied by the squared norm of the imbedding curvature tensor of Multiply CR-warped product statistical submanifolds N of holomorphic statistical space forms M. Furthermore, we prove that under certain geometric conditions, N and M become Einstein.

Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form

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application/pdf Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form Michel Nguiffo Boyom, Jamali Mohammed, Hasan Shahid

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In this article, we derive an inequality satisfied by the squared norm of the imbedding curvature tensor of Multiply CR-warped product statistical submanifolds N of holomorphic statistical space forms M. Furthermore, we prove that under certain geometric conditions, N and M become Einstein.

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Multiply CR- Warped Product Statistical Submanifolds of a Holomorphic Statistical Space From By PROF. MOHAMMAD HASAN SHAHID Department of Mathematics Faculty of Natural Sciences Jamia Millia Islamia(Central University) New Delhi, India (with Prof. Michal Boyom and M. Jamali) CR-submanifold and CR-warped Product Submanifold Let B and F be two Riemannian Manifolds with Riemannian metric and , respectively , and f a positive differentiable function on B. The warped product manifold equipped with the Riemannian metric The function f is called the warping function. It is well known that the notion of warped product plays some important roles id differential geometry as well as in physics. Bg Fg FB´ FB gfgg 2 += Let M be a Kaehler manifold with complex structure J and N a Riemannian manifold isometrically immersed in . For each , we denote by the maximal holomorphic subspace of the tangent space of N. If the dimension of is the same for all , the space define a holomorphic distribution D on N, which is called the holomorphic distribution of N. A submanifold N is a Kaehler manifold is called a CR-submanifold if there exists a holomorphic distribution D on N whose orthogonal complement is totally real distribution, i.e., . A CR-submanifold is called a totally real submanifold if dim =0. M NxÎ xD NTx xD NxÎ xD M ^ D NTJD ^^ Ì xD Statistical manifolds introduced, in 1985, by Amari have been studied in term of information geometry. Since the geometry of such manifolds includes the notion of dual connections, also called conjugate connection in affine geometry, it is closely related to the affine differential geometry. Further, a statistical structure being a generalization of a Hessian geometry. Let be Riemannian manifold and M a submanifold of . If is a statistical manifold, then we call a statistical submanifold of , where is an affine connection on M and g is the metric tensor on M induced from the Riemannian metric on . Let be an affine connection on . If is astatistical manifold and M a submanifold of , then is also a statistical manifold by induced connection and metric g. ),( gM M ),,( gM Ñ ),,( gM Ñ ),( gM Ñ g M Ñ M ),,( ÑgM M ),,( gM Ñ Ñ In the case is a semi-Riemannian manifold, the induced metric connection g has to be non-degenerated. In the geometry of submanifolds, Gauss formula, Weingarten formula and the equation of Gauss, Codazzi and Ricci are know as fundamental equations. Corresponding fundamental equations on statistical submanifolds were obtained . Let M be an n-dimensional submanifold of . Then, for any , Gauss formula is ),( gM M )(, TMYX GÎ ),( YXhYY XX +Ñ=Ñ ),( YXhYY XX *** +Ñ=Ñ Where h and are symmetric and bilinear, called the imbedding curvature tensor of M in for and the imbedding curvature tensor of M in for , respectively. it is also proved that and are dual statistical structure on M, where g is induced metric on from the Riemannian metric on . Let us denote the normal bundle on M by . Since h and are bilinear, we have the linear transformation and defined by * h M Ñ M * Ñ ),( gÑ ),( g* Ñ )(TMG g M )( ^ G TM * h xA * xA )),,((),( xx YXhgYXAg = )),,((),( xx YXhgYXAg ** = Definition. Let be Riemannian manifold of the dimensions respectively and let be the Cartesian product of . For each a, denote by the canonical projection N and . We denote the horizontal lift of in N via by itself. If are positive valued functions, then (2.1) define a metric g on N. The product manifold N endowed with this metric is denoted by . This product manifold N is known as multiply warped product manifold. kNNN ,....,, 21 knnn ,....,, 21 kNNNN ,....,, 21= kNNN ,....,, 21 aa NN®:p aN aN ap aN + ®RNk 12 :,.....,ss YXYXYXg aa k a a ** 1 2 1*1*1 ,)(,),( pppspp å= += o kk NNN ss ´´´ .....221 Definition. If be k statistical manifolds, then N= is again a statistical manifold with metric given by equation (2.1). This manifold N is called multiply warped product statistical manifold. Now let us denote the part by and by . Then N can be represented as . We denote by as the vector field on M and X, Y…. the induced vector field on N. Definition. A multiply warped product statistical submanifold in an almost complex manifold M is called a multiply CR-warped product statistical submanifold if is an invariant submanifold and is an anti-invariant submanifold of M. kNNN ,....,, 21 kk NNN ss ´´´ .....221 kk NN ss ´´...22 ^N 1N TN ^´= NNN T )(...., MYX GÎ ^´= NNN T TN ^N We denote by , m ≥ 1 the Euclidean 2m space with the standard metric. Then the canonical complex structure of is defined by (*) Example. Consider in the submanifold is given by the equations [B. Sahin, Geom. Dedicata 2006] ),,...,,(),,...,,( 1111 mmmm xyxyyxyxJ --= From (*) one can obtain that TM is spanned by , where Using (*) one gets that is invariant with respect to J. Moreover, are orthogonal to TM. Hence, is anti-invariant with respect to J. Thus M is a CR-submanifold of . Furthermore, we can derive that and are integralable. Denoting the integral manifold of D and by , respectively, then the induced metric tensor is Thus M is a CR-warped product submanifold of with warping function . • A. Bejancu, CR- submanifold of a Kaehler a manifold I, Proc. Amer. Math. Soc. 69 (1978), 135-142. • A. Bejancu, CR-submanifold of a Kaehler manifold II, Trans. Amer. Math. Soc. 69 (1979), 333-345. • Chen BY (1981) CR-submanifolds in Kaehler manifolds. I. J Diff Geometry 16: 305-322; CR-submanifolds in Kaehler manifolds. II. Ibid 16: 493-509. • Chen BY (2001) Geometry of warped product CR- submanifolds in Kaehler manifolds I. Monatsh Math 133: 177-195; Geometry of warped product CR-submanifolds in Kaehler manifolds. II. Ibid 134: 103-119. • S . Amari, Differential-Geometrical methods in Statistics, Springer-Verlag, 1985. • Yano K. and Kon, M.: CR-submanifolds of Kaehlerian and Sasakian Manifolds, Birkhauser, Basel, 1983. From the decomposition of and we may write Also for multiply CR- warped product statistical submanifolds N of a statistical manifold [L. Tod., Diff. Geom. – Dynamical system ,2006] and (4) for any vector fields and , where denotes the - component of Z . ),(),(),( yxhYXhYXh JD l+= ^ a k a a zXz ))(log( 2 å= = s a k a aX ZXZ ))(log( 2 å= * =Ñ s DX Î ^ ÎDZ a Z aN ^ Å= DDTN lÅ= ^^ JDNT Lemma 1. Let be a multiply CR- warped product statistical submanifold of a holomorphic statistical space form M. then we have (i) (ii) (iii) For any vector field X in D and Z , W in , where denotes the - component of Z. JXJPJZXYJXh z a a k a JD += å= ^ ))(log(),( 2 s ),(),( JWJXQgWJXPg ZZ = )),(,(),()),(),,(( 2 ZXJhXQgXZhZXJhZJXhg Z ll += ^ D a Z aN Proof. From Gauss formula we can write (5) where P and Q denotes the tangential and normal projection. ),(),( ZXJhXJXQXPYJXhJX ZZZZ +Ñ++=+Ñ ),())(log((),( 2 XZJhZXJXQXPZJXh a k a aZZ +++= å= s Comparing the tangential part in the above equation and then taking inner product with , we get^ Î DW ^ = ÎÎ"+=å^ DZDXJXJPZXZJXh Z a k a aJD ,,))(log(),( 2 s å= - k a a a JZJX 2 ))(log( s Now comparing normal parts of (5) and taking inner product with JW for Using part(i) of the lemma 1 we arrive at ^ ÎDW )),()(log(),()),,(( 2 JWJZgXJWXQgJWZJXhg a k a aZJD å= +^ s ).,(),( JWXQgWJXPg ZZ = Comparing normal part of on both the sides and taking inner product with we find Theorem 2. Let be multiply CR-warped product statistical submanifold of holomorphic statistical space form M with , then the square norm of imbedding curvature tensor of N in M satisfies the following inequalities : DDPD Î^ å= +=- k a a aZ JZXXQXZJhZJXh 2 )(log(),(),( sl Proof. Let be local orthonormal frame of vector field frame of the vector field on and be such that is a basis for some , a= 2,…..,k where ,…., and },...,,,....,,{ 21121 pppp JXJXJXXXXX ==+ TN },...,,{ 21 qZZZ a Z D aN },...,2,1{ 22 n=D }...,...,1....{ 21132 kkk nnnnnn +++++++=D - qnnn k =+++ .....32 The above equation implies Now using part (i) of the Lemma 1 we get In the view of the above assumption , the above inequality takes the form DDPD Î^ By the Cuachy-Schwartz inequality the above equation becomes Therefore Theorem 3. Let be a compact orientable multiply CR- warped product statistical submanifold without boundary of holomorphic statistical space form M of constant curvature k. If and Then And the equality holds if and if Proof. Let , , then form holomorphic statistical space form of constant curvature k, we have DDPD Î^ DXÎ ^ ÎDZ Which implies . (7) On the other hand from Codazzi equation, we may write (8) Now, we calculate each term of (8) as (10) Similarly we replace X By JX in the last equation, we get (11) (12) (13) (14) . (16) å= += k a aa aa ZZgXXXJZZJXXR 2 2 )],(})log()log([{),,,( ss å= * ++Ñ- k a aa aaX ZZgJXJXJXJZZJXhg 2 2 )],(})log()log([{)),,(( ss å= * Ñ-Ñ- k a aa aXJX ZZgXJZZJXhg 2 ),()log()),,(( s å= Ñ- k a aa aJX ZZgJX 2 )},()log{( s Combining (7) and (16) and taking summation over the range from 1 to p, we have (18) 2 2 2 2 )(log) 4 ( a k a a k a a ZZ pk åå == D= s å= - k a aDgrad 2 2 )(log s å= *^*^ Ñ-Ñ+ p i Jeiei JZZehgJZZJehg ii 1 )]),,(()),,(([ Integrating both the sides, Green’s and the hypothesis leads to Since And Further the equality holds if and only if Which implies that the equality holds if . This proves the theorem. 0 })(log{4 2 2 2 22 £ - = å ò å ò = = k a N a k a N aD a dvZp dvgradZ k s 0 2 2 >òå= N k a a dvZp 0})(log{ 2 2 2 ³òå= dvgradZ N aD k a a s 0})(log{ 2 =ò dvgrad N aD s Theorem 4. Let be a compact orientable anti-invariant multiply warped product statistical submanifold without boundary of holomorphic statistical space form M of constant curvature k. If and , then and the equality holds if and only if Proof. From the previous theorem we have . Since N is anti-invariant , we have and . DDPD Î^ XAJZA JXJZ JXX * ^ * ÑÑ = ),(),,,( * ³ HHgYXYXR 0)(log =aDgrad s 0=TN ^= NN This implies that N becomes completely totally umbilical submanifold of M. Furthermore, from the expression of the ambient curvature we have, for two orthonormal vector Then . Furthermore, from Gauss equation and totally umbilicity of N, we obtain and the equality holds if TNYX Î, 4 ),,,( k YXYXR -= )),( 4 (),,,( *+-= HHg k YXYXR ),(),,,( * ³ HHgYXYXR Theorem 5. Let be a compact orientable anti-invariant multiply warped product statistical submanifold without boundary of holomorphic statistical space form M of constant curvature k. If and , then M is Einstein and N is Einstein if and only if is constant. Proof. The proof is straight from the last theorem and the Gauss equation which combinely give DDPD Î^ XAJXA JZJZ JXX ^ * ^ * ÑÑ = ),( 4 * + HHg k ),()},( 4 ){1(),( ZYgHHg k nZYRic * +-= References: [1]. S. AMARI, “Differential Geometric methods in statistics”, Springer-Verlag, 1985. [2]. S. AMARI and H. NAGAOKA, “Methods of Information Geometry”, Transl. Math. Monogr., Vol-191, Amer. Math. Soc., 2000. [3]. M. E. AYDIN, A. MIHAI and I. MIHAI, “Some inequalities on submanifolds in statistical manifolds of constant curvature”, Filomat (To appear). [4]. R.L. BISHOP and B. O’NEILL, “Manifolds of negative curvature”, Trans. of Amer. Math. Soc., Vol-145(1969), 1-49. [5]. B. Y. CHEN, “Geometry of warped product CR-submanifolds in Kaehler manifold”, Monatsh. Math., 133(2001), 177-195. [6]. B. Y. CHEN, “Geometry of warped product CR-submanifolds in Kaehler manifold II”, Monatsh. Math., 134(2001), 103-119. [7]. B. Y. CHEN and FRANKI DILLEN, “Optimal inequalities fir multiply warped product submanifolds”, Int. Elect. J. of Geometry, Vol-1 (2008), No-1, 1-11. [8]. H. FURUHATA, “Hypersurfaces in statistical manifolds”, Diff. Geom. Appl., 27, (2009), 420-429. [9]. L. Todgihounde, “Dualistic structures on warped product manifolds”, Differential Geometry-Dynamical Systems, Vol-8 (2006), 278-284. [10]. P. W. VOS, “Fundamental equations for statistical submanifolds with applications to the Bartlett connection”, Ann. Inst. Statist. Math., 41(3) (1989), 429-450.