**INTRODUCTION**

The elastic curve is one of the important topics in geometry that appear in
computation of the solution of a variational problems which for the first time
was considered by Daniel Bernoulli and Leonhard Euler in 1744. The main goal
of their study was minimizing the bending energy of a thin inextensible wire.
Some additional information about their works is available in work of Love
(1927), Hsu (2007), Yanti and
Mahlia (2009), Akanmu and Gambo (2007, 2008).
The major idea of this variational problem is minimizing the energy function
defined as the integral of the squared curvature for a curve of a fixed length
subject to boundary conditions (Barros *et al*., 1999).

According to Koiso (1992), for a curve in a Riemannian
manifold, we can define two quantities: the length and the total square curvature
of curve. Also a curve is called as an elastica if it is a critical point of
the functional total square curvature restricted to the space of curves of a
fixed length.

One of essential work in elastica is classified the all closed elastic curves
in the Euclidean space, (Langer and Singer, 1984). Also,
Koiso (1992) found the unique long time solution of
an initial value problem in Euclidean space and studied the elastic curves restricted
in a submanifold. Barros *et al*. (1999) studied
the complete classification of elastic curves in complex projective plane.

Singer (2007), studied the elastica in Euclidean space
and classified the elastic curves in a Riemannian manifold with constant sectional
curvature G. The cross sectional curvature G = 0 has been studied by Sager
* et al*. (2011). The aim of this paper was to study the classical
variational problem in the 3-dimensional indefinite-Riemannian manifold.

**PRELIMINARIES**

Let M_{γ} be a 3-dimensional indefinite-Riemannian manifold of
index γ (0≤γ≤3) isometrically immersed into an m-dimensional
indefinite-Riemannian manifold
of index i for m≥3. Then is M_{γ} called 3-dimensional indefinite-Riemannian
submanifold of M_{i}. Especially if γ = 1, then M_{1} is
called a Lorentzian submanifold of M_{i} (Lopez,
2008). We denote the metrics of M_{γ} and
by the symbol 〈,〉 and the covariant differentiation of M_{γ}
(resp. )
by ∇ (resp.).
Then we have the Gauss formula:

where, X and Y are tangent vector fields of M_{γ} and B is the second fundamental form of M_{γ}.

Let α(t) be a regular curve on a 3-dimensional indefinite-Riemannian manifold M_{γ}. We denote the tangent vector field α’(t) = X. When 〈X, Y〉 = ±1, α is called a unit speed curve.

**THE ELASTICA IN A 3-DIMENSIONAL INDEFINITE-RIEMANNIAN MANIFOLD**

This study has formulated a generalized variational problem, that of the elastica
in 3-dimensional indefinite-Riemannian manifold. By this we mean a curve which
is an extremal for the integral of the squared (geodesic) curvature among curves
with specified boundary conditions. Here, we summarize the machinery needed
for calculations.

In what follows, M is a smooth 3-dimensional indefinite-Riemannian manifold, with indefinite-Riemannian metric g (X, Y) = 〈X, Y〉 = x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}, where X = (x_{1}, x_{2}, x_{3}) and Y = (y_{1}, y_{2}, y_{3}), that is, a symmetric bilinear form on tangent vectors X and Y at each point. The covariant derivative ∇_{X}Y, measures the derivative of a vector field Y in the direction of a vector X.

**Definition:** A vector field V is called spacelike if 〈V,V〉>0
or V = 0, timelike if 〈V,V〉<0 and lightlike if 〈V,V〉
= 0 and V≠0.

For vector fields X and Y the equality of mixed partial derivatives is replaced
by the bracket formula:

Let α(t) be an immersed curve in M, then it has velocity vector V = vT
and squared geodesic curvature:

Set the Frenet frame for a family of curves α_{w}(t) = g(w,t) by (T, N, B), therefore we can write:

where, V is velocity,
is speed, W represents an infinitesimal variation of the curve and s is the
arc-length parameter along a curve.

The basic formulas needed in calculating the Euler equations are as follows:

So:

So:

Here, the curvature tensor R is given by:

The proof of Eq. 6, is mentioned in Singer
(2007).

In what follows, α[0,1]→M is a curve of length L. Now for fixed constant λ let:

For a variation α_{W} with variation field W we compute:

One of the symmetries of the curvature tensor allows us to replace 〈R(W, T)T, ∇_{T}T〉 with 〈R(∇_{T}T, T)T, W〉. Now integrate by parts, using g = -〈∇_{T}W, T〉, we get:

where:

and:

**THE FRENET EQUATIONS**

Let α be a curve in 3-dimensional indefinite-Riemannian manifold M with
speed v(t) = |α’(t)|, curvature k, torsion τ and Frenet frame
{T, N, B}. The Frenet equations are written down as follows (Fernandez
*et al*., 2006):

where, ε_{1} is 〈T, T〉, ε_{2} is 〈N, N〉 and ε_{3} is 〈B, B〉.

**The timelike case:** Let α(t) be a timelike curve in 3-dimensional indefinite-Riemannian manifold M. If the normal vector field N and the binormal vector field B are spacelike, then we have the following Frenet formulas along α(t):

where, κ and τ are curvature and torsion of α, respectively.

**Theorem 1:** Let α(t) be a timelike curve in the smooth 3-dimensional
indefinite-Riemannian manifold M, then α(t) is the elastic curve if and
only if the curvature κ, torsion τ and sectional curvature G of α
being as follows:

where, c_{1}, c_{2} are constants and:

**Proof:** Since M is a manifold of sectional curvature G, the formula for E can be simplified to:

Then:

The equations E = 0 for the elastica become:

Solving above system, we get:

where, c_{1}, c_{2} are constants and:

Conversely, by substituting the κ, τ and G in the Eq. 6, we can get E = 0, therefore, α is an elastic curve.

**The spacelike case:** Let α(t) be a spacelike curve in M. There are three possibilities depending on the causal character of ∇_{T}T, which are given, respectively, in the following theorems:

**Theorem 2:** Let α(t) be a spacelike elastic curve in the smooth
3-dimensional indefinite-Riemannian manifold M with a constant sectional curvature
G and covariant differentiation ∇. Suppose that the vector field ∇_{T}T
be a spacelike, then the curvature is
and the torsion is
where c, m, p, q and r are constants and s is arc-length.

**Proof:** The Frenet equations for α are as follows:

where, κ and τ are curvature and torsion of α, respectively. In this case, the formula for E can be simplified to:

where, s is the arc-length.

The equations E = 0 for the elastica become:

The second equation integrates to:

where, c is constant. Multiplication of the first equation by 2κ_{s} and integration yields:

where, A is undetermined constant. Letting u = κ^{2}, this becomes:

Since, this equation is of the form u^{2}_{s} = p (u) and P
a third degree polynomial, it can be solved by standard techniques in terms
of elliptic functions (Jing-Lei and Zhi-Jian, 2011;
Abazari, 2011; Koklu, 2002).
The cubic polynomial P (u) satisfies P (0) = 4c^{2}≥0 and lim_{u→±∞}
P (u) = ±∞. Furthermore, if u = κ^{2} is a nonconstant solution
to (9), it must obviously take on values at which P (u)>0. It follows that
we may assume P(u) has three real roots, given in two cases:

**Case I** |
**:** |
-α_{1},-α_{2},α_{3},
satisfying -α_{1}≤α_{2}≤0≤α_{3} |

**Case II** |
**:** |
α_{1},α_{2},α_{3} satisfying 0≤α_{1}≤α_{2}≤α_{3} |

We can now write Eq. 9 in the form:

The solution of Eq. 10 is given by:

where:

For background on the solution of such equations, (Davis,
1962). Also, α_{1}, α_{2} and α_{3}
are related to the coefficients of P(u) by:

where, the up and down symbol are related to case I and case II, respectively.

**Theorem 3:** Let α(t) be a spacelike curve in the smooth 3-dimensional
indefinite-Riemannian manifold M with a covariant differentiation ∇. Suppose
that the vector field ∇_{T}T be a timelike, then α(t) is
the elastic curve if and only if the curvature κ, torsion τ and sectional
curvature G of α is as follows:

where, c_{1}, c_{2} are constants and:

**Proof:** The Frenet equations for α are as follows:

where, κ and τ are curvature and torsion of α, respectively. In this case, the formula for E can be simplified to:

The equations E= 0 for the elastica become:

Integrates of first and third equation yields:

Therefore, G = c where:

Conversely, by substituting the κ, τ and G in the Eq. 15, we can get E = 0, therefore, α is an elastic curve.

**Theorem 4:** Let α(t) be a spacelike curve in the smooth 3-dimensional indefinite-Riemannian manifold M with a constant sectional curvature G and covariant differentiation ∇. Suppose that the vector field ∇_{T}T be a lightlike, then the curvature is κ = 1 and the torsion is:

where:

and c is integration constant.

**Proof:** The vector field ∇_{T}T is lightlike. Then, similar on previous case, the Frenet equations for α are as follows:

where, κ = 1 and τ is torsion of α. In this case, the formula for E can be simplified to:

The equations E = 0 for the elastica become:

or:

where:

Since, d is a constant then the general solution of Eq. 18 is obtained as Eq. 16.

**ACKNOWLEDGMENTS**

The authors would like to thank Prof. Dr. H.H. Hacisalihoglu for his careful reading of the first draft and many helpful suggestions. Also the first author would like to thanks the Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran, for its financial support.