IJST (2011) A4: 265-271 Iranian Journal of Science & Technology http://www.shirazu.ac.ir/en Tangent Bishop spherical images of a biharmonic B-slant helix in the Heisenberg group Heis 3 T. Korpinar, E. Turhan* and V. Asil Department of Mathematics, Firat University, 23119, Elazıg, Turkey E-mails: talatkorpinar@gmail.com, essin.turhan@gmail.com, vasil@firat.edu.tr Abstract In this paper, biharmonic slant helices are studied according to Bishop frame in the Heisenberg group Heis 3 . We give necessary and sufficient conditions for slant helices to be biharmonic. The biharmonic slant helices are characterized in terms of Bishop frame in the Heisenberg group Heis 3 . We give some characterizations for tangent Bishop spherical images of B-slant helix. Additionally, we illustrate four figures of our main theorem. Keywords: Biharmonic curve; Bishop frame; Heisenberg group; tangent Bishop spherical images 1. Introduction Let ) , ( g M and ) , ( h N be manifolds and N M : a smooth map. Denote by the connection of the vector bundle TN induced from the Levi-Civita connection h of ) , ( h N . The second fundamental form d is defined by . , , = , TM Y X Y d Y d Y X d X X Here is the Levi-Civita connection of ) , ( g M . The tension field is a section of TN defined by . = d tr (1) A smooth map is said to be harmonic if its tension field vanishes. It is well known that is harmonic if and only if is a critical point of the energy: g dv d d h E , 2 1 = over every compact region of M . Now let N M : be a harmonic map. Then the Hessian H of E is given by *Corresponding author Received: 1 November 2010 / Accepted: 18 April 2011 . , , , = , TN W V dv W V h W V g J H Here the Jacobi operator J is defined by , , := TN V V V V R J (2) , , = , := 1 = 1 = i i N m i i e i e i e i e m i e d e d V R V R (3) where N R and i e are the Riemannian curvature of N , and a local orthonormal frame field of M , respectively [1]. Let ) , ( ) , ( : h N g M be a smooth map between two Lorentzian manifolds. The bienergy ) ( 2 E of over compact domain M is defined by . , = 2 g dv h E A smooth map ) , ( ) , ( : h N g M is said to be biharmonic if it is a critical point of the ) ( 2 E , [2-11]. The section ) ( 2 is called the bitension field of and the Euler-Lagrange equation of 2 E is 0. = ) ( := ) ( 2 J (4) In [12] the authors completely classified the biharmonic submanifolds of codimension greater than one in the n-dimensional sphere. The