Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms

07/11/2017
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22639
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit
 

Résumé

Statistical manifolds are abstract generalizations of statistical models introduced by Amari [1] in 1985. Such manifolds have been studied in terms of information geometry which includes the notion of dual connections, called conjugate connection in affine geometry. Recently, Furuhata [5] de ned and studied the properties of holomorphic statistical space forms.
In this paper, we obtain the generalized Wintgen type inequality for Lagrangian submanifolds in holomorphic statistical space forms. We also obtain condition under which the submanifold becomes minimal or H is some scalar multiple of H*.

Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms

Collection

application/pdf Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms Michel Nguiffo Boyom, Mohd Aquib, Mohammad Hasan Shahid, Mohammed Jamali
Détails de l'article
contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms

Média

Voir la vidéo

Métriques

0
0
228.97 Ko
 application/pdf
bitcache://eb50f8492ad09a9137a64f7b339b053cef50bdb2

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors Platine

alanturinginstitutelogo.png
logothales.jpg

Sponsors Bronze

logo_enac-bleuok.jpg
imag150x185_couleur_rvb.jpg

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors

smai.png
logo_gdr-mia.png
gdr_geosto_logo.png
gdr-isis.png
logo-minesparistech.jpg
logo_x.jpeg
springer-logo.png
logo-psl.png

Organisateurs

logo_see.gif
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/17410/22639</identifier><creators><creator><creatorName>Michel Nguiffo Boyom</creatorName></creator><creator><creatorName>Mohammad Hasan Shahid</creatorName></creator><creator><creatorName>Mohd Aquib</creatorName></creator><creator><creatorName>Mohammed Jamali</creatorName></creator></creators><titles>
            <title>Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2018</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Wintgen inequality</subject><subject>Lagrangian submanifold</subject><subject>holomorphic statistical space forms</subject></subjects><dates>
	    <date dateType="Created">Fri 9 Mar 2018</date>
	    <date dateType="Updated">Fri 9 Mar 2018</date>
            <date dateType="Submitted">Sat 21 Apr 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">eb50f8492ad09a9137a64f7b339b053cef50bdb2</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>37396</version>
        <descriptions>
            <description descriptionType="Abstract">Statistical manifolds are abstract generalizations of statistical models introduced by Amari [1] in 1985. Such manifolds have been studied in terms of information geometry which includes the notion of dual connections, called conjugate connection in affine geometry. Recently, Furuhata [5] de ned and studied the properties of holomorphic statistical space forms.<br />
In this paper, we obtain the generalized Wintgen type inequality for Lagrangian submanifolds in holomorphic statistical space forms. We also obtain condition under which the submanifold becomes minimal or H is some scalar multiple of H*.
</description>
        </descriptions>
    </resource>
.

Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms Michel Nguiffo Boyom1 , Mohd. Aquib2 , Mohammad Hasan Shahid3 , and Mohammed Jamali4 1 Nguiffo Boyom M. IMAG: Alexander Grothendieck Research Institute, Universit of Montpellier France nguiffo.boyom@gmail.com, 2 Department of Mathematics, Jamia Millia Islamia University, New Delhi, India aquib80@gmail.com, 3 Department of Mathematics, Jamia Millia Islamia University, New Delhi, India hasan jmi@yahoo.com, 4 epartment De Mathematiques, Al-Falah University, Haryana, India jamali dbd@yahoo.co.in Abstract. Statistical manifolds are abstract generalizations of statisti- cal models introduced by Amari [1] in 1985. Such manifolds have been studied in terms of information geometry which includes the notion of dual connections, called conjugate connection in affine geometry. Re- cently, Furuhata [5] defined and studied the properties of holomorphic statistical space forms. In this paper, we obtain the generalized Wintgen type inequality for Lagrangian submanifolds in holomorphic statistical space forms. We also obtain condition under which the submanifold becomes minimal or H is some scalar multiple of H∗ . Keywords: Wintgen inequality, Lagrangian submanifold, holomorphic statistical space forms 1 Introduction The history of statistical manifold was started from investigations of geometric structures on sets of certain probability distributions. In fact, statistical mani- folds introduced, in 1985, by Amari [1] have been studied in terms of informa- tion geometry and such manifolds include the notion of dual connections, called conjugate connection in affine geometry, closely related to affine differential ge- ometry and which has application in various fields of science and engineering such as string theory, robot control, digital signal processing etc. The geometry of submanifolds of statistical manifolds is still a young geometry, therefore it attracts our attention. Moreover, the Wintgen inequality is a sharp geometric inequality for surface in 4-dimensional Euclidean space involving Gauss curvature (intrinsic invariant), normal curvature and square mean curvature (extrinsic invariant). The gener- alized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for the submanifolds in real space forms also known as DDVV conjecture. In present article, we will prove the generalized Wintgen type inequalities for Lagrangian submanifolds in statistical holomorphic space forms some of its applications. 2 Statistical manifolds and submanifolds A statistical manifold is a Riemannian manifold (M, g) endowed with a pair of torsion-free affine connections ∇ and ∇ ∗ satisfying Zg(X, Y ) = g(∇ZX, Y ) + g(X, ∇ ∗ ZY ), (1) for X, Y, Z ∈ Γ(TM). It is denoted by (M, g, ∇, ∇ ∗ ). The connections ∇ and ∇ ∗ are called dual connections and it is easily shown that (∇ ∗ )∗ = ∇. The pair (∇, g) is said to be a statistical structure. If (∇, g) is a statistical structure on M, then (∇ ∗ , g) is also statistical structure on M. Denote by R and R ∗ the curvature tensor fields of ∇ and ∇ ∗ , respectively. Then the curvature tensor fields R and R ∗ satisfies g(R ∗ (X, Y )Z, W) = −g(Z, R(X, Y )W). (2) Let M be a 2m-dimensional manifold and let M be a n-dimensional submanifolds of M. Then, the corresponding Gauss formulas according to [7] are: ∇XY = ∇XY + h(X, Y ) (3) ∇ ∗ XY = ∇∗ XY + h∗ (X, Y ) (4) where h and h∗ are symmetric and bilinear, called imbedding curvature tensor of M in M for ∇ and the imbedding curvature tensor of M in M for ∇ ∗ , respectively. Let us denote the normal bundle of M by Γ(TM⊥ ). Since h and h∗ are bilinear, we have the linear transformations Aξ and A∗ ξ defined by g(AξX, Y ) = g(h(X, Y ), ξ), (5) g(A∗ ξX, Y ) = g(h∗ (X, Y ), ξ), (6) for any ξ ∈ Γ(TM⊥ ) and X, Y ∈ Γ(TM). The corresponding Weingarten for- mulas [7] are: ∇Xξ = −A∗ ξX + ∇⊥ Xξ, (7) ∇ ∗ Xξ = −AξX + ∇∗⊥ X ξ, (8) for any ξ ∈ Γ(TM⊥ ) and X ∈ Γ(TM). The connections ∇⊥ X and ∇∗⊥ X given in the above equations are Riemannian dual connections with respect to the induced metric on Γ(TM⊥ ). The corresponding Gauss, Codazzi and Ricci equations are given by the fol- lowing results. Proposition 1 ([7]) Let ∇ be a dual connection on M and ∇ the induced con- nection on M. Let R and R be the Riemannian curvature tensors of ∇ and ∇, respectively. Then, g(R(X, Y )Z, W) = g(R(X, Y )Z, W) + g(h(X, Z), h∗ (Y, W)) − g(h∗ (X, W), h(Y, Z)), (9) (R(X, Y )Z)⊥ = ∇⊥ Xh(Y, Z) − h(∇XY, Z) − h(Y, ∇XZ) − {∇⊥ Y h(X, Z) − h(∇Y X, Z) − h(X, ∇Y Z)}, (10) g(R⊥ (X, y)ξ, η) = g(R(X, y)ξ, η) + g([A∗ ξ, Aη]X, Y ), (11) where R⊥ is the Riemannian curvature tensor on TM⊥ , ξ, η ∈ Γ(TM⊥ ) and [A∗ ξ, Aη] = A∗ ξAη − AηA∗ ξ. Similarly, for the dual connection ∇ ∗ on M, we have Proposition 2 ([7]) Let ∇ ∗ be a dual connection on M and ∇∗ the induced connection on M. Let R ∗ and R∗ be the Riemannian curvature tensors of ∇ ∗ and ∇∗ , respectively. Then, g(R ∗ (X, Y )Z, W) = g(R∗ (X, Y )Z, W) + g(h∗ (X, Z), h(Y, W)) − g(h(X, W), h∗ (Y, Z)), (12) (R ∗ (X, Y )Z)⊥ = ∇∗⊥ X h∗ (Y, Z) − h∗ (∇∗ XY, Z) − h∗ (Y, ∇∗ XZ) − {∇∗⊥ Y h∗ (X, Z) − h∗ (∇∗ Y X, Z) − h∗ (X, ∇∗ Y Z)}, (13) g(R∗⊥ (X, y)ξ, η) = g(R ∗ (X, y)ξ, η) + g([Aξ, A∗ η]X, Y ), (14) where R∗⊥ is the Riemannian curvature tensor for ∇⊥∗ on TM⊥ , ξ, η ∈ Γ(TM⊥ ) and [Aξ, A∗ η] = AξA∗ η − A∗ ηAξ. Definition 1 ([5]). A 2m-dimensional statistical manifold M is said to be a holomorphic statistical manifold if it admits an endomorphism over the tangent bundle Γ(M) and a metric g and a fundamental form ω given by ω(X, Y ) = g(X, JY ) such that J2 = −Id; ∇ω = 0, (15) for any vector fields X, Y ∈ Γ(M). Since ω is skew-symmetric, we have g(X, JY ) = −g(JX, Y ). Definition 2 ([5]). A holomorphic statistical manifold M is said to be of con- stant holomorphic curvature c ∈ R if the following curvature equation holds : R(X, Y )Z = c 4 {g(Y, Z)X − g(X, Z)Y + g(X, JZ)JY −g(Y, JZ)JX + 2g(X, JY )JZ}. (16) According to the behavior of the tangent space under the action of J, subman- ifolds in a Hermitian manifold is divided into two fundamental classes namely: Invariant submanifold and totally real submanifold. Definition 3. A totally real submanifold of maximal dimension is called La- grangian submanifold. Let {e1, . . . , en} and {en+1, . . . , e2m} be tangent orthonormal frame and nor- mal orthonormal frame, respectively, on M. The mean curvature vector field is given by H = 1 n n X i=1 h(ei, ei) (17) and H∗ = 1 n n X i=1 h∗ (ei, ei). (18) We also set khk2 = n X i,j=1 g(h(ei, ej), h(ei, ej)) (19) and kh∗ k2 = n X i,j=1 g(h∗ (ei, ej), h∗ (ei, ej)). (20) 3 Generalized Wintgen type inequality We denote by K and R⊥ the sectional curvature function and the normal curva- ture tensor on M, respectively. Then the normalized scalar curvature ρ is given by [7] ρ = 2τ n(n − 1) = 2 n(n − 1) X 1≤i