On the energy of frenet vectors fields in Rn

Abstract: In this paper, we compute the energy of Frenet vector fields for a given curve C in n-dimensional Euclidean space. We observe that the energy and angle may be expressed in terms of the curvature functions of C. If the first curvature function of the curve is the identity function then its first integral is the angle between the velocity vectors and its second integral gives the energy of the velocity vector of the curve.


Introduction
The volume of unit vector fields has been studied by Gluck and Ziller (1986), Johnson (1988), and Higuchi, Kay, and Wood (2001) among other scientists. They define the volume of unit vector field X as the volume of the submanifold of the unit tangent bundle defined by X(M) . In Wood (1997), the energy of a unit vector field on a Riemannian manifold M is defined as the energy of the mapping X : M → T 1 M, where the unit tangent bundle T 1 M is equipped with the restriction of the Sasaki metric.
Generally, every geometric problem about curves can be solved using the curves' Frenet vectors field. Therefore, in this paper, we focus on the curve C, instead of the manifold M. Let C be a curve with a pair (I, ) of parametric unit speed in R n . Let us take an initial point a ∈ I and the Frenet frames {V 1 ( (a)), … , V r ( (a))} and {V 1 ( (s)), … , V r ( (s))} at the points (a) and (s), respectively. We calculated the energy of a Frenet vector field and, in Altın (1999), the angle between each vector V i ( (a)) and V i ( (s)) where 1 ≤ i ≤ r. Further, we observed that the energy and angle may be expressed in terms of the curvature functions of the given curve C. If the 1st curvature function of the curve is identity function its first integral is angle between vectors V 1 ( (a)) and V 1 ( (s)), its second integral is the energy, from a to s, of the velocity vector of curve. Then we defined the energy of the velocity and velocity field of the curve. This definition will give us a new approach to elastic curves (see Brander, Gravesen, & Nrbjerg, 2017;Guven , Valencia, & Vazquez-Montejo, 2014;Santiago, Chacón-Acosta, & Gonzalez-Gaxiola, 2013

ABOUT THE AUTHOR
Ayşe Altın is an associate professor. She received her PhD in Differential Geometry from Ankara University, Ankara, Turkey. Nowadays, she works as the lecturer and researcher at Department of Mathematics, Hacettepe University, Ankara, Turkey. She is an expert in differential geometry.

PUBLIC INTEREST STATEMENT
In this study we calculated the energy of the Frenet vector fields of a curve and found the energy of the velocity vector field as half of the integral of the square of the first curvature function.
We expect that this result will give us a new approach to Classical Bernoulli-Euler Elastic Curves. Definition 1.1 Let I be an open interval in R and be a differentiable map : I → R n . We call (I) = C a curve C in R n and (I, ) a parametric pair for C.
Theorem 1.1 Let (I, ) be a parametric pair for a curve C in a space R n . There exists a parametric pair (J, ) of the curve C such that for each r ∈ J, ‖ � (r)‖ = 1, where J is an open interval in R. The pair (J, ) is called a parametric pair with unit speeds (O'Neill, 1966).
Definition 1.2 Let (I, ) be a parametric pair of a curve C in a space R n . Let the system Ψ = { � , �� , … , r } be a maximal linearly independent set. The orthonormal system {V 1 , V 2 , … , V r } obtained from Ψ is named as Frenet frame fields of C, and {V 1 ( (s)), V 2 ( (s)), … , V r ( (s))} at the point (s) ∈ C as Frenet frames.
Definition 1.3 Let (I, ) be a parametric pair for a curve C in a space R n and {V 1 ( (s)), V 2 ( (s)), … , V r ( (s))} be Frenet frames at the point (s) ∈ C. Let be defined as curvature function on C and the real number k i ( (s)) be defined as ith curvature on C at the point (s).
Theorem 1.2 Let (I, ) be a parametric pair for a curve C in a space R n . If we take ith curvature k i ( (s)) and Frenet frames {V 1 ( (s)), V 2 ( (s)), … , V r ( (s))} at the point (s), then, the following relations are hold: (2) For ∈T x M and a section : M → T 1 M, we have: where ∇ is the Levi-Civita covariant derivative (Chacón, Naveira, & Weston, 2001).
This gives a Riemannian metric on TM. As mentioned, g  is called the Sasaki metric. The metric g s makes the projection : T 1 M → M a Riemannian submersion (Chacón et al., 2001).
where is the canonical volume form in M and {e a } is a local basis of the tangent space (see Chacón & Naveira, 2004;Wood, 1997).
The energy of a unit vector field X is defined to be the energy of the section X: M → T 1 M, where T 1 M is the unit tangent bundle equipped with the restriction of the Sasaki metric on TM. Now let :T 1 M → M be the bundle projection, and let T(T 1 M) =  ⊕  denote the vertical/horizontal splitting induced by the Levi-Civita connection. Further, define TM =  ⊕ where  denotes the line bundle generated by X, and  is the orthogonal complement (Chacón et al., 2001). Furthermore, the energy of the velocity vector fields of the curve is related to the elastic curves (elastica) (see Brander et al., 2017;Guven et al., 2014;Santiago et al., 2013 for examples).

The energy of Frenet vectors fields
Now, we are in a position to prove our main result which was pointed out before.
Theorem 2.1 Let C be a curve with a pair (I, ) of parametric unit speeds in R n . Let us take an initial point a ∈ I. Further, let be the Frenet frames at the points (a) and (s), respectively. Then we have the following conditions: {V 1 ( (a)), … , V r ( (a))} and {V 1 ( (s)), … , V r ( (s))} . Proof (i) Let TC be the tangent bundle and let {V 1 , V 2 , … , V r } be Frenet vector fields of the curve C. So we have V 1 : C → TC = ⋃ t∈I T (t) C. Let : TC → C be the bundle projection. The Levi-Civita connection map K: T(TC) → TC. By using Equation (5) we obtain the energy of V 1 as where du is the element of arc length. From (4) we have On the other hand, by Proposition 1.1, we may write Then we obtain From (1) we get By putting (7) in (6), we get Let N i C be the ith normal bundle. Thus we have V i : C → N i C where N i C = ⋃ t∈I N i (t) C and here N i (t) C denotes generated by V i . Now, let i : N i C → C be the i th bundle projection. The Levi-Civita connection map K i : T(N i C) → N i C By using Equation (5), we obtain the energy of V i , 2 ≤ i ≤ r as From (4) we have By (2) we have by using (8), we obtain So, (3) gives us and then (8) yields We may ignore the constant term of 1 2 (s − a) and we can give the following definition.

Definition 2.1 The integral
is called the energy of the velocity vector field of curve C at a fixed point a ∈ I, and is denoted by (V 1 ).

Conclusion
In this work, we calculate the energy of the Frenet vectors fields and the angle between the vectors V i (a) and V i (s), where 1 ≤ i ≤ r . So, we see that both energy and angle depend on the curvature functions of the curve C and the energy of velocity vector field is (V 1 (s)) = 1 2 � s a k 2 1 (u)du + 1 2 (s − a).
We may ignore the constant term of 1 2 (s − a) and we can give the definition. The integral (V 1 )(s) = 1 2 � s a k 2 1 (u)du is called the energy of the velocity vector field of curve C at a fixed point a ∈ I.
On the other hand, the classical curve known as the elastica is the solution to a variational problem proposed by Daniel Bernoulli to Leonhard Euler (1744), that of minimizing the bending energy of a thin inextensible wire (See, e.g. Love, 1927). The mathematical idealization of this problem is that of minimizing the integral of the squared curvature for curves of a fixed length satisfying given firstorder boundary data (Singer, 2007). It is obvious, the energy of the velocity vector fields of the curve is related to the elastic curves.