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On Geodesic Behavior of Some Special Curves S S symmetry Article On Geodesic Behavior of Some Special Curves Savin Trean¸t˘a Department of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania; [email protected] Received: 2 March 2020; Accepted: 17 March 2020; Published: 1 April 2020 Abstract: In this paper, geometric structures on an open subset D ⊆ R2 are investigated such that the graphs associated with the solutions of some special functions to become geodesics. More precisely, we determine the Riemannian metric g such that Bessel (Hermite, harmonic oscillator, Legendre and Chebyshev) ordinary differential equation (ODE) is identified with the geodesic ODEs produced by the Riemannian metric g. The technique is based on the Lagrangian (the energy of the curve) 1 L = k x˙(t) k2, the associated Euler–Lagrange ODEs and their identification with the considered 2 special ODEs. Keywords: auto-parallel curve; geodesic; Euler–Lagrange equations; Lagrangian; special functions MSC: 34A26; 53B15; 53C22 1. Introduction and Preliminaries The concept of connection plays an important role in geometry and, depending on what sort of data one wants to transport along some trajectories, a variety of kinds of connections have been introduced in modern geometry. Crampin et al. [1], in a certain vector bundle, described the construction of a linear connection associated with a second-order differential equation field and, moreover, the corresponding curvature was computed. Ermakov [2] established that linear second-order equations with variable coefficients can be completely integrated only in very rare cases. Further, some aspects of time-dependent second-order differential equations and Berwald-type connections have been studied, with remarkable results, by Sarlet and Mestdag [3]. Michor and Mumford [4] considered some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S1 to the plane modulo the group of diffeomorphisms of S1, acting as reparameterizations. For an excellent survey on geometric dynamics, convex functions and optimization methods on Riemannian manifolds, the reader is directed to Udri¸ste[5,6]. Relatively recently, Udri¸steet al. [7], by using an identity theorem for ordinary differential equations (ODEs), investigated some geometrical structures that transform the solutions of a second order ODE into auto-parallel graphs. Later, Trean¸t˘aand Udri¸ste[8], in accordance to Udri¸steet al. [7], studied the auto-parallel behavior of some special plane or space curves by using the theory of identifying of two ODEs. In the present paper, as a natural continuation of some results obtained in Trean¸t˘aand Udri¸ste[8], we are looking for an appropriate geometric structure such that important graphs in applications (like Bessel functions, Hermite functions etc) to become geodesics. Specifically, our aim is to determine the Riemannian metric gij such that Bessel ordinary differential equation, Hermite ODE, harmonic oscillator ODE, Legendre ODE and Chebyshev ODE, respectively, is identified with the geodesic 1 ODEs produced by g . The technique is based on the Lagrangian (the energy of the curve) L = k ij 2 x˙(t) k2, the associated Euler–Lagrange ODEs and their identification to the Bessel ODE, Hermite Symmetry 2020, 12, 504; doi:10.3390/sym12040504 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 504 2 of 11 ODE, harmonic oscillator ODE, Legendre ODE and Chebyshev ODE, respectively. By applying this new technique, we developed an original point of view by introducing some new results regarding the geodesics curves literature. For this aim, in the following, we present some basics to be used in the sequel. Let M be a differentiable manifold of dimension n and denote by X (M) the Lie algebra of vector fields on M.A Riemannian metric on M is a family of (positive definite) inner products gp : Tp M × Tp M ! R, p 2 M, such that for any two differentiable vector fields X, Y 2 X (M), the application p ! gp (X(p), Y(p)) defines a smooth function M ! R. i j ¶ ¶ The local expression of a Riemannian metric g is g = gijdx ⊗ dx , where gij = g , , i, j 2 ¶xi ¶xj f1, ..., ng, determine at every point p 2 M a symmetric positive definite matrix. The inverse of the −1 ij ¶ ¶ ij tensor field g is g = g ⊗ , where g are the entries of the inverse matrix of (gij). Endowed ¶xi ¶xj with this metric, the differentiable manifold (M, g) is called Riemannian manifold. Let (M, g) be a Riemannian manifold and gij the components of g. The Riemannian metric g determines the symmetric 1 ¶g ¶gjl ¶gjk linear connection Gi = gil kl + − , i, j, k 2 f1, 2, ..., ng (Christoffel symbols), which is jk 2 ¶xj ¶xk ¶xl ¶gij called the Riemannian (Levi–Civita) connection of M, whose fundamental property is r g = − k ij ¶xk l l gljGik − gliGjk = 0 (the tensor field g is parallel with respect to the symmetric linear connection r). Let (M, g) be a C¥-class n-dimensional Riemannian manifold and r a linear connection on M. We say that a vector field X 2 X (M) is a parallel vector field with respect to r if and only if rY X = 0, for all Y 2 X (M). Consider c : [a, b] ! M, c(t) = x1(t), ..., xn(t) , t 2 [a, b] ⊆ R, a regular differentiable curve on M. We say that a vector field X 2 X (M) is a parallel vector field along the curve c with respect to r if rc˙(t)X = 0. The curve c is called auto-parallel if rc˙(t)c˙(t) = 0 or, equivalently, it satisfies d2xi dxj dxk + Gi = 0, i = 1, n. dt2 jk dt dt The paper is structured as follows. Section2 includes the main results of the present paper. First, the geodesic behavior of some special curves on an open subset D ⊆ R2 is analyzed. In the second part of this section, the general case is investigated and, in this way, the results obtained in the first part of Section2 become non-trivial illustrative examples of the developed theory. Finally, Section3 contains conclusions and other development ideas. 2. Main Results In this section, we formulate and prove the main results of the paper. First, we study the geodesic behavior of some special curves on an open subset D ⊆ R2. According to the previous section, a C2-class curve x : I ⊆ R ! D, x(t) = x1(t), x2(t) = (t, y(t)), is called auto-parallel with respect to the symmetric linear connection r on D, of components i Gjk, i, j, k 2 f1, 2g, if and only if rx˙(t)x˙(t) = 0 or, d2xi dxj dxk + Gi = 0, i, j, k 2 f1, 2g, dt2 jk dt dt or, equivalently, the function y : I ! R is a solution of the following differential equations 1 2 1 1 G22(t, y)y˙ + 2G12(t, y)y˙ + G11(t, y) = 0 (1) 2 2 2 2 y¨ + G22(t, y)y˙ + 2G12(t, y)y˙ + G11(t, y) = 0. (2) i Let assume that the previous symmetric linear connection r on D, of components Gjk, i, j, k 2 f1, 2g, is the Levi–Civita connection, that is, its components fulfill the following differential equations Symmetry 2020, 12, 504 3 of 11 ¶g ¶gjl ¶gjk 2g Gi = kl + − , i, j, k, l 2 f1, 2g , (3) il jk ¶xj ¶xk ¶xl i j where gij[= gji], i, j 2 f1, 2g, represent the components of the Riemannian metric g = gijdx ⊗ dx on D. Taking into account the complete integrability conditions (closeness conditions) for the components 1 i i of the Riemannian metric g = gij on D, and also, Gjk = 0 and Gjk = Gkj for i, j, k 2 f1, 2g, we can rewrite the differential Equation (3) as follows ¶g ¶g ¶g 2g G2 = 11 , 2g G2 = 22 , 2g G2 = 11 (4) 21 11 ¶t 22 22 ¶y 21 12 ¶y ¶g ¶g ¶g ¶g ¶g 2g G2 = 2 12 − 11 , 2g G2 = 22 , 2g G2 = 2 12 − 22 ; 22 11 ¶t ¶y 22 21 ¶t 21 22 ¶y ¶t ¶g ¶G2 ¶g ¶G2 12 G2 + g 11 = 12 G2 + g 12 (closeness conditions) ¶y 11 12 ¶y ¶t 12 12 ¶t ¶g ¶G2 ¶g ¶G2 22 G2 + g 12 = 22 G2 + g 22 ¶y 12 22 ¶y ¶t 22 22 ¶t ¶g ¶G2 ¶g ¶G2 ¶g ¶G2 ¶g ¶G2 22 G2 + g 11 + 12 G2 + g 12 = 12 G2 + g 22 + 22 G2 + g 12 . ¶y 11 22 ¶y ¶y 12 12 ¶y ¶t 22 12 ¶t ¶t 12 22 ¶t 2.1. Bessel Geodesics We begin by recalling Bessel ODE, 1 a2 y¨(x) + y˙(x) + 1 − y(x) = 0. x x2 According to Proposition 3.1 in Trean¸t˘aand Udri¸ste[8], we have 1 1 1 1 G22 = 0, G12 = G21 = 0, G11 = 0 a2 1 G2 = 1 − y, G2 = G2 = , G2 = 0. 11 t2 12 21 2t 22 ¶g Therefore, using Equation (4) (see the second equation), we obtain 22 = 0, that is, g = j(t). ¶y 22 Also, using the sixth equation in Equation (4), we get ¶g ¶g 1 1 2 12 = j˙ (t) () 12 = j˙ (t) () g = g = j˙ (t)y + y(t). ¶y ¶y 2 12 21 2 ¶g From the third equation, 2g G2 = 11 , it follows 21 12 ¶y ¶g j˙ (t)y + 2y(t) j˙ (t) y(t) 11 = () g = y2 + y + k(t). ¶y 2t 11 4t t Moreover, putting the condition that the above mentioned components of the Riemannian metric i j g = gijdx ⊗ dx on D to satisfy the remaining equations in Equation (4), we find 2y(t)y˙ (t) y2(t) j(t) = tc, c > 0; k˙(t) = − tc t2c Symmetry 2020, 12, 504 4 of 11 0 1 y(t) a2 c a2 − t2 = 0, c 4a2 − 4t2 − 1 = 0, − 1 − y(t) = 0.
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