Biharmonic submanifolds in 3dimensional (κ, µ)-manifolds

Biharmonic maps between Riemannian manifolds are defined as critical points of the bienergy and generalized harmonic maps. In this paper, we give necessary and sufficient conditions for nonharmonic Legendre curves and anti-invariant surfaces of 3-dimensional (κ,μ)-manifolds to be biharmonic.


Introduction
Let f : (M,g) → (N,h) be a smooth map between two Riemannian manifolds.The bienergy E 2 ( f ) of f over compact domain Ω ⊂ M is defined by where τ( f ) is the tension field of f and dv g is the volume form of M.
It is clear that E 2 ( f | Ω ) = 0 on any compact domain if and only if f is a harmonic map.Thus E 2 provides a measure for the extent to which f fails to be harmonic.If f is a critical point of (1.1) over every compact domain, then f is called a biharmonic map or 2-harmonic maps.Jiang [10] proved that f is biharmonic if and only if here f is the Jacobi operator of f .Clearly, any harmonic map is biharmonic.But the converse is not true.Nonharmonic biharmonic maps are said to be proper.It is well known that proper biharmonic maps into R, that is, biharmonic functions, play an important role in elasticity and hydrodynamics.
Proper biharmonic submanifolds in real space forms have been studied by many geometers during the last two decades.However, in the Euclidean space and the hyperbolic space, such submanifolds have not been found yet.On the other hand, many proper biharmonic submanifolds exist in the unit sphere.
The unit sphere of odd dimension is the typical example of Sasakian space forms.Lately, J. Inoguchi and T. Sasahara initiated the study of proper biharmonic submanifolds in Sasakian space forms.Inoguchi [9] classified proper biharmonic Legendre curves and Hopf cylinders (automatically anti-invariant surfaces) in Sasakian 3-space forms.Sasahara [11] classified proper biharmonic Legendre surfaces in Sasakian 5-space forms.Also, in [1], the authors studied anti-invariant submanifolds in Sasakian 5-space forms.
In this paper, in terms of the curvature and the torsion, (resp., the mean curvature), we give necessary and sufficient conditions for nonharmonic Legendre curves (resp., antiinvariant surfaces) in 3-dimensional (κ,µ)-manifolds to be biharmonic.

(κ,µ)-manifolds
In this section, we collect some basic facts about contact metric manifolds.We refer to [3] for a more detailed treatment.A (2n + 1)-dimensional differentiable manifold N 2n+1 is called a contact manifold if there exists a globally defined 1-form η such that (dη) n ∧ η = 0. On a contact manifold there exists a unique global vector field ξ satisfying for all X ∈ TN 2n+1 .Moreover, it is well known that there exist a (1,1)-tensor field φ and a Riemannian metric g which satisfy for all X,Y ∈ TN 2n+1 .

Biharmonic maps
Let M m and N n be Riemannian manifolds and f : M m → N n a smooth map.The tension field τ( f ) of f is a section of the vector bundle f * TN n defined by where ∇ f , ∇, and {e i } denote an induced connection, the Levi-Civita connection of M m , and a local orthonormal frame field of M m , respectively.A smooth map f is said to be a harmonic map if its tension field vanishes.It is well known that f is harmonic if and only if f is a critical point of the energy: over every compact domain Ω of M m .Eells and Sampson [8] suggested to study biharmonic maps which are critical points of the bienergy E 2 : The Euler-Lagrange equation of the functional E 2 was computed by Jiang [10] as follows: Here the operator f is the Jacobi operator defined by where R N n is the curvature tensor of N n .

Biharmonic Legendre curves
A curve C = C(s) : I → M 3 (κ,µ) parametrized by arclength parameter is said to be a Legendre curve if η(C ) = 0.In this section, in terms of the curvature and the torsion, we characterize proper biharmonic Legendre curves in 3-dimensional (κ,µ)-manifolds.
The tangent planes in T p N 2n+1 which is invariant under φ are called φ-section (see [3]).The sectional curvature of φ-section is called φ-sectional curvature.If the φ-sectional curvature is constant on N 2n+1 , then N 2n+1 is said to be of constant φ-sectional curvature.Complete and connected Sasakian manifolds of constant φ-sectional curvature are called Sasakian space forms.Denote Sasakian space forms of constant φ-sectional curvature c by N 2n+1 (c).

Biharmonic anti-invariant surfaces
Let M m be a submanifold tangent to ξ in a contact metric manifold.If φX is normal to TM m for any X ∈ TM m , then M m is called an anti-invariant submanifold (see [12]).
Let f : M 2 → M 3 (k,µ) be a nonminimal anti-invariant surface.The formulas of Gauss and Weingarten are given, respectively, by where X,Y ∈ TM m , V ∈ T ⊥ M m , σ, A, and D are the second fundamental form, the shape operator, and the normal connection.Denote by R the Riemann curvature tensor of M 2 .Then the equations of Gauss and Codazzi are given, respectively, by where X, Y , Z, W are vectors tangent to M 2 , •, • = g(•,•) and ∇σ is defined by (5.4) Let {e 1 ,e 2 } be orthonormal frame fields along M 2 such that e 2 = ξ.We may assume that H = αφe 1 , where α ∈ C ∞ (M) and α > 0. Then from (2.6), we see that the second fundamental form σ takes the following form: A φe1 e 2 = −βe 1 . (5.9) We put γ = he 1 ,φe 1 .We need the following lemmas for the later use.
Using Lemma 5.4, we obtain the following system of partial differential equations.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Corollary 4 . 5 .
Let C : I → M 3 (c) be a Legendre curve in Sasakian space forms of constant φ-sectional curvature c.Then C is proper biharmonic if and only if c > 1 and C is a helix satisfying α 2 = c − 1.