In this paper, we discuss some geometric properties of Riemannian submersions whose fibres are totally contact umbilical. Some interrelations between totally contact umbilic, totally geodesic and minimality are established

Tshikuna-Matamba, T.

English

T. TSHIKUNA-MATAMBA

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

A Note on Riemannian Submersions with Umbilical Fibres

Scitech Research Journals

Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   778  SCITECH                                                                         Volume 6, Issue 2  RESEARCH ORGANISATION|                Published online: January 13, 2016| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals     A Note on Riemannian Submersions with Umbilical Fibres T. TSHIKUNA-MATAMBA D partement de Math matiques Institut Sup rieur P dagogique de Kananga B.P. 282-Kananga R publique D mocratique du Congo. tshikmat@gmail.com  ABSTRACT.  In this paper, we discuss some geometric properties of Riemannian submersions whose fibres are totally contact umbilical. Some interrelations between totally contact umbilic, totally geodesic and minimality are established. 2000 Mathematics Subject Classification: 53C15, 53C25. Key words and phrases: Riemannian submersions; almost contact metric manifolds; umbilicity; almost contact metric submersions.  1. INTRODUCTION      Riemannian submersions have been initiated by B. O’Neill in [6], but various classes are treated in [3] and the references therein. The case of totally umbilic fibres was introduced by G. Baditoiu and S. Ianus in [1].       In this note, we will be interested by almost contact metric submersions which are Riemannian submersions whose total space is an almost contact metric manifold studied by B. Watson [10]. The paper focuses on the geometry of the fibres, emphasing on the concept of totally contact umbilicity. It is organized in the following way.       In Section 2, we recall some basic notions related to almost contact metric manifolds.      Section 3 is devoted to almost contact metric submersions. Relationships between this concept with the minimality or the geodesibility are established. Furtheremore, we apply, to the study of almost contact metric submersions of type II, a result of Hong and Tripathi [4] concerning the asymptotic direction generated by the characteristic vector field of the total space. 2. BACKGROUND ON MANIFOLDS Let be a differentiable manifold of odd dimension  An almost contact structure on  is a triple  where: (1)  is a characteristic vector field, Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   779 (2)  is a 1−form such that  and (3)  is a tensor field of type (1, 1) satisfying  where   is the identity transformation. If  is equipped with a Riemannian metric   then   is called an almost contact metric structure. So, the quintuple  is an almost contact metric manifold. Any almost contact metric manifold admits a fundamental 2−form,  defined by  An almost contact metric manifold is said to be: (1) closely cosymplectic if  and  (2) nearly cosymplectic if   (3) nearly-K- cosymplectic if  and  (4) nearly Kenmotsu if  and  if  and  (5) K-contact We begin by recalling some basic concepts before applying them to the case of Riemannian submersions.        Let   be a submanifold of a Riemannian manifold  and  the second fundamental form of  . It is well known that the mean curvature vector field, H, of  is given by where  denotes the trace of  and  the dimension of .    If  then  is said to be minimal;    If  then  is said to be totally geodesic, and    If  then  is totally umbilical.    In the case of contact geometry, M.M. Tripathi and S.S. Shukla, [7], have defined the concepts of totally contact umbilic and totally contact geodesic in the following way. Let  be an almost contact metric manifold. For a distribution  on ,,   is said to be  totally geodesic if, for all  we have     If for all we have for some normal vector field   then is  totally umbilical.    Suppose that  is tangent to the structure vector field  Consider the distribution generated by this vector field and  its complement. In this case, Tripathi and Shukla have defined the following two concepts. The submanifold  is said to be: Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   780 (1) totally contact umbilic if it is −totally umbilic; (2) totally contact geodesic if it is  −totally geodesic. Definition 2.1. Let be a submanifold of an almost contact metric manifold     Then  is: (a) totally contact umbilic if,                    for all   and a unit normal vector field. (b) totally contact geodesic if,  for all    In his study of lightlike hypersurfaces of indefinite Sasakian manifolds, F. Massamba [5] used also these two concepts. Proposition 2.2. Let  be a submanifold of an almost contact metric manifold  if  is:     (1) totally contact umbilic, then,      (2) totally contact geodesic, then,  Proof. (1) Let us consider the relation  It is known that  from which   On the other hand,                Remembering that  we get the proof of assertion (1). Concerning assertion (2), since  is totally contact geodesic, we have which implies that  But  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   781 which is   Proposition 2.3. ([4]) Let  be a submanifold of an almost contact metric manifold tangent to  If is totally contact umbilic or totally contact geodesic, then  is an asymptotic direction. Proof. Let us recall that a vector field  defines an asymptotic direction if  With this in mind, putting   in the defining equations in Proposition 2.2, it is clear that the structure vector field  defines an asymptotic direction.                                                                       3. ALMOST CONTACT METRIC SUBMERSIONS In [6], O’Neill has defined a Riemannian submersion as a surjective mapping  between two Riemannian manifolds such that   (i)    is of maximal rank;  (ii)  is a linear isometry. The tangent bundle  of the total space  admits an orthogonal decomposition  where  and  denote respectively the vertical and horizontal distributions. We denote by and  the vertical and horizontal projections respectively. A vector field  of the horizontal distribution is called a basic vector field if it is        related to a vector field  of the base space   Such a vector field means that        On the base space, tensors and other objects will be denoted by a prime’ while those tangent to the fibres will be specified by a carretˆ. Herein, vector fields tangent to the fibres will be denoted by   and        When the base space is an almost Hermitian manifold,  the Riemannian submersion  is called an almost contact metric submersion of type II, [8], if       This type of submersions are called  holomorphic in [2].     Now, let us turn to some implications of the total umbilicity. Proposition 3.1. Let  be a Riemannian submersion with totally umbilic fibres. If the mean curvature vector field,  of the fibres is parallel,   Proof. Assertions (a) and (c) follow from O’Neill’s equations [6]. Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   782       Let us consider (b). Recall from Baditoiu and Ianus [1], that     Since,   is vertical and  is horizontal, we have  on the other hand, the parallelism of implies that  so that  therefore        Let us examine some applications in the case of almost contact metric submersions.     Let  and be two almost contact metric manifolds. By an almost contact metric submersion of type I, in the sense of Watson [10], one understands a Riemannian submersion                                       satisfying      When the base space is an almost Hermitian manifold,  the Riemannian submersion                                       is called an almost contact metric submersion of type II, if [10].     Now, we overview some of the fundamental properties of these submersions. Proposition 3.2. Let be an almost contact metric submersion of type I. Then (a)  (b)  (c)  for all   (d) is the basic vector field associated to if  and are basic. Proof. See [8, 9, 10].  Proposition 3.3. Let an almost contact metric submersion of type II. Then (a)  (b)   for all  (c) is the basic vector field associated to  if  and are basic. Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   783 Proof. See again Watson [10].                                                                                                                     Proposition 3.4. Let be an almost contact metric submersion of type I. If the fibres are totally contact umbilic, then                           Proof. It is known that the O’Neill configuration tensor  is the second fundamental form of the fibres. Since  vanishes on vertical vector fields, then in Proposition 2.2, the defining equation (a) becomes which is  because                  Corollary 3.5. Let be an almost contact metric submersion of type I with totally contact umbilic fibres. If the fibres are minimal, then they are totally geodesic. Proof. In such a case, we have                           because of umbilicity of the fibres. If moreover the fibres are minimal, we have   which shows that                                                                                                                                                         Proposition 3.6. Let be an almost contact metric submersion of type II. Suppose that the total space  is a  contact manifold. If the fibres are totally contact umbilic, then they are totally geodesic. Proof. See [4].                                                                                                                                      It is easy to show that if   then With this in mind, we can deduce that         The fibres of an almost contact metric submersion whose total space is closely cosymplectic, nearly cosymplectic, nearly-  - cosymplectic or nearly Kenmotsu are totally geodesic. Proposition 3.7. Let  be an almost contact metric submersion of type II. If the fibres are totally contact umbilic, then the vector fields defines an asymptotic direction.. Proof. Since, for a type II submersion,  is tangent to the fibres, then we can apply Proposition 2.3.          Note. Let  be a Riemannian submersion whose total space is a warped product. In this case, we have a submersion with totally umbilic fibres as noted in [3]. Since a Kenmotsu manifold is a warped product, [3], it is clear that any almost contact metric submersion with total space a Kenmotsu manifold is of totally umbilic fibres.  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            ISSN: 2395-0218        Volume 6, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm                                                   784 References [1] BADITOIU, G. and IANUS, S., Semi-Riemannian submersions with totally umbilic fibres, Rend. Circ. Mat.Palermo, Series II, 51 (2002), 249-276. [2] BUREL, J. M., Almost contact structures and harmonic maps with minimal fibres, Houston J. Math., 30(2) (2004), 393-411. [3] FALCITELLI, M., IANUS, S. and PASTORE, A.M., ”Riemannian submersions and related topics”, World Sci. Pub.Co. 2004. [4] HONG, S. and TRIPATHI, M.M., Ricci curvature of submanifolds of a Sasakian space form, Iranian J. Math. Sc. Info., 1(2) (2006),31-52. [5] MASSAMBA, F., Totally contact umbilical lightlike hypersurfaces of indefinite Sasakian manifolds, Kodai Math.J., 31 (2008), 338-358. [6] O’NEILL, B., The fundamental equations of a submersion, Michigan Math.J., 13 (1966), 459-469. [7] TRIPATHI, M.M. and SHUKLA, S.S., Semi-invariant submanifolds of nearly Kenmotsu manifolds, Bull. Cal. Math. Soc., 95(1) (2003), 17-30. [8] TSHIKUNA-MATAMBA, T., A note on G1−almost contact metric submersions, Bull. Cal. Math. Soc., 103(3)(2011),255-264. [9] TSHIKUNA-MATAMBA, T., A note on almost contact metric submersions whose total space is a Chinea-Gonzalez manifold, Ann. Univ. Bucharest, Mathematical Series., 3(LXI)(2012),111-122. [10] WATSON B.,The differential geometry of two types of almost contact metric submersions, in The Math.Heritage of C.F. Gauss, (Ed.G.M. Rassias), World Sci. Publ. Co. Singapore, 1991, pp. 827-861.   

http://scitecresearch.com/journals/index.php/jprm/article/download/545/387

A Note on Riemannian Submersions with Umbilical Fibres

Abstract

Similar works

Full text

Available Versions

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Scitech Research Journals