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The geodesics of a pseudo-Riemannian manifold.


According to [12], [13], a pseudo-Riemannian metric of signature (p, q) on a smooth manifold M of dimension n = p + q is a smooth symmetric differentiable 2-form g on M such that, at each point x of M, [g.sub.x] is non-degenerate on [T.sub.x]M with the signature (p, q). We call (M, g) a pseudo-Riemannian manifold.

Let be given a pseudo-Riemannian manifold (M, g). The fundamental theorem of pseudo-Riemannian geometry states that there exists an unique linear connection [[nabla].sub.g] on M, called the Levi-Civita connection (of g), such that the following two assertions hold good:

a) [[nabla].sub.g] is metric (i.e. [[nabla].sub.g] g = 0); b) [[nabla].sub.g] is torsion-free (i.e. T = 0).

If (U, [x.sup.1], ..., [x.sup.n]) is a coordinate chart on M, then the Christoffel symbols [[GAMMA].sup.k.sub.ij] of the Levi-Civita connection are related to the functions [g.sub.ij] by the formulas


Also, the curvature R has the components [R.sup.l.sub.ijk] given by


Note that in local coordinates a geodesic [gamma](t) = ([x.sup.i](t))i=1, ..., n satisfies a system of n second order differential equations


If f: M [right arrow] R is a smooth function, then the second covariant derivative

[[nabla].sup.2.sub.g] f = ([[partial derivative].sup.2] f/[partial derivative][x.sup.i][partial derivative][x.up.j] - [[GAMMA].sup.k.sub.ij] [partial derivative]f/[partial derivative][x.sup.k]) [dx.sup.i] [cross product] [dx.sup.j]

is called the Hessian of f, [14], [17].

Let us suppose that the Hessian h = [[nabla].sup.2.sub.g] f is non-degenerate. Then h is a pseudo-Riemannian metric which produces the Levi-Civita connection [[nabla].sub.h] and the Christoffel symbols [bar.[GAMMA].sup.k.sub.ij].

Throughout this paper, we shall use the following notations:


We have, [3]

Theorem 1.1. Let [f,] be the contravariant components of the pseudo-Riemannian metric [] = [f.sub.,pk] and [R.sup.m.sub.ijk] be the components of the curvature tensor field produced by the pseudo-Riemannian metric [g.sub.ij]. Then the components of Levi-Civita connection [[nabla].sub.h] are given by the following formula

[[GAMMA].sup.p.sub.ij] = [[GAMMA].sup.p.sub.ij] + 1/2 [f,] [absolute value of [f,.sub.ijk] + ([R.sup.m.sub.ijk] + [R.sup.m.sub.jki]) f, m].

Corollary 1.1. The differential system of geodesies is

[[??].sup.p] + [[[GAMMA].sup.p.sub.ij] + [f,] (1/2 [f,.sub.ijk] + [R.sup.l.sub.ikj] f,l)] [[??].sup.i] [[??].sup.j] = 0, p = 1, ..., n.

We remark that Corollary 1.1 is the generalization of Theorem 2.1 from [9] in the pseudo-Riemannian case.


Let us take M = [R.sup.2.sub.+] be the positive quadrant endowed with the metric g = ([g.sub.ij]), where

[g.sub.ij](x, y) = diag (1/[x.sup.2], 1/[y.sup.2]).

It is known that this metric has the Christoffel coefficients

[[GAMMA].sup.1.sub.11] = - 1/x, [[GAMMA].sup.2.sub.22] = - 1/y, [[GAMMA].sup.2.sub.11] = [[GAMMA].sup.1.sub.21] = [[GAMMA].sup.1.sub.12] = [[GAMMA].sup.2.sub.12] = [[GAMMA].sup.2.sub.21] = [[GAMMA].sup.1.sub.22] = 0.

We choose the function

f: [R.sup.2.sup.+] [right arrow] R, f(x, y) = 1/x + [square root of x] + 1/y [square root of y],

and we prove that [[nabla].sup.2] f = ([f.sub.,ij]) is positive definite on (M, g).

Indeed, after straightforward calculation, we get

[[nabla].sup.2] f = diag (1/[x.sup.3] + 1/4x [square root of x], 1/[y.sup.3] + 1/4y [square root of y])

which is positive definite on [R.sup.2.sub.+], therefore ([R.sup.2.sub.+], [[nabla].sup.2] f) is a Riemannian manifold.

According to Theorem 1.1, we have

[bar.[GAMMA].sup.1.sub.11] = 3/x (1/4 + 1/4 + 1/4 + x [square root of x]), [bar.[GAMMA].sup.2.sub.22] = 3/y (1/4 + 1/4 + 1/4 + y [square root of y]),

[bar.[GAMMA].sup.2.sub.11] = 0, [bar.[GAMMA].sup.1.sub.12] = [bar.[GAMMA].sup.1.sub.21] = 0, [bar.[GAMMA].sup.2.sub.12] = [bar.[GAMMA].sup.2.sub.21] = 0, [bar.[GAMMA].sup.1.sub.22] = 0,

and with p = 1 in Corollary 1.1, we find x"(t) = - [bar.[GAMMA].sup.1.sub.11][(x'(t)).sup.2]. We put x = x(t) and state

Theorem 2.1. The component x of geodesic curves of the Riemannian manifold ([R.sup.2.sub.+], [[nabla].sup.2] f) is solution of the 2nd order ODE

x" = 3/x (1/4 + 1/4 + x [square root of c])[(x').sup.2], (2-1)

with arbitrary initial values.

Remark 2.1. With p = 2 in Corollary 1.1, we find y"(t) = [bar.[GAMMA].sup.2.sub.22][(y'{t)).sup.2], therefore y satisfies the ODE in Theorem 2.1 too.

To find a more convenient form of equation (2.1), we arrange it in the form

x"/x' = 3 x'/4 x + 3x'/x (4 + x [square root of x]);

and, after an integration, in the form

ln x'(t) = 3/4 ln x(t) + 3 [integral] x'(t)/x(t) (4 + x(t) [square root of x(t)]) dt. (2.2)

We denote by I the integral in the right side of (2.2). Changing the variable [square root x] = u reduces this integral to the form

I = 2 [integral] 1/u(4 + [u.sup.3]) du.

Using the general procedure for the integration of rational functions, after straightforward calculation, we obtain a primitive

I = ln [square root of x]/6 [square root of x] [square root of x] + 4 + ln [k.sub.1],

where [k.sub.1] is any positive real constant. Using this primitive, we can write (2.2) as

ln x'(t) = 3/4 ln x(t) + 3 ln 4 [square root of x(t)]/6 [square root of x(t)] [square root of x(t)] + 4 + ln [k.sub.1]. (23)

After an arrangement in (2.3) and Rising for y the same form, we can state our main result. This is given in

Theorem 2.2. The geodesic curves of the Riemannian manifold ([R.sup.2.sub.+], [[nabla].sup.2] f) are solutions bf the [first order ODE system

x'(t) = [[k.sub.1]x(t).sup.3/2]/[[square root of x(t)].sup.3/2] + 4, y'(t) = [[k.sub.2]y(t).sup.3/2]/[[square root of y(t)].sup.3/2] + 4 (2.4)

where [k.sub.1] fond [k.sub.2] are any positive real constants, and the initial values are arbitrary.

The equations (2.4) cannot he integrated using Analytical methods because we meet a Chebyshev integral of this form [integral] [u.sup.-2] [square root of 3] [+ or -] 4 du. However, a numerical study of this system could be useful.


In this section, we find an implicit parametric solution of (2.4), and use a numerical method to produce the direction geodesic held plots as well as a phase portrait bf solution curves.

For our first purpose, it is enough to make the study of the first equation in (2.4). We consider both x and [k.sub.1] be positive, and we write the first equation in (2.4) as

[k.sub.1] dt = [square root of x] [square root of x + 4]/x [square root of x] dx. (3.1)

If we integrate (3.1) side by side, we obtain

[k.sub.1]t + [c.sub.1] = [integral] [square root of x] [square root of x + 4]/x [square root of x] dx.

Changing the variable [square root of x] = v reduces this equality to the form

[k.sub.1]t + [c.sub.1] = 2 [integral] [square root of [v.sup.3]] + 4 [v.sup.2] dv.

or, more convenient, to the form

[k.sub.1]t + [c.sub.1] = [integral] (v + 4/[v.sup.2]). 1/[square root of 1 + [v.sup.3]/4]] dv. (3.2)

If v [member of] (0, 3 [square root 4]), then we can make use of Taylor expansion and write (3.2) as

[k.sub.1]t + [c.sub.1] = [integral] ((v + 4/[v.sup.2]) [summation over n [greater than or equal to] 0] [(-1).sup.n] (2n)!/[2.sup.2n][(n!).sup.2][([v.sup.3]/4).sup.n]) dv. (3.3)

If we commute the sum and the integral in (3.3), after integration we obtain

[k.sub.1]t + [c.sub.1] = [summation over (n [greater than or equal to] 0)] [(-1).sup.n] (2n)!/[2.sup.4n][(n!).sup.2][([v.sup.3n + 2]/3n + 2 + [4v.sup.3n - 1]/3n - 1).

Finally, we return to x as variable. So, if x [member of] (0, 3 [square root of 16]), we have

[k.sub.1]t + [c.sub.1] = [summation over (n [greater than or equal to] 0)] [(-1).sup.n] (2n)!/[2.sup.4n][(n!).sup.2][([x.sup.1/2 (3n + 2)]/3n + 2 + [4x.sup.1/2 (3n - 1)]/3n - 1). (3.4)

We underly that equation (3.4) gives an implicit parametric form of the solution x = x(t, [k.sub.1], [c.sub.1]) of the ODEs in Theorem 2.2. A similar form can be written for y = y (t, [k.sub.2], [c.sub.2]), where [k.sub.2] and [c.sub.2] are real constants, [k.sub.2] > 0. Therefore, we have obtained this result.

Theorem 3.1. Let 0 < x, y < 3 [square root of l6]. The implicit parametric solution of ODEs in Theorem 2.2 is given by

[k.sub.1]t + [c.sub.1] = [summation over (n>0)][(-1).sup.n] (2n)!/[2.sup.4n][(n!).sup.2] ([x.sup.1/2(3n + 2)]/3n + 2)/3n + 2 + [4x.sup.1/2(3n - 1])/3n - 1);

[k.sub.2]t + [c.sub.2] = [summation over (n [greater than or equal to] 0)][(-1).sup.n] (2n)!/[2.sup.4n][(n!).sup.2] ([y.sup.1/2(3n + 2)/3n + 2)]/3n + 2 + [4y.sup.1/2(3n - 1])/3n - 1),

[k.sub.1]; [c.sub.1], [k.sub.2] and [c.sub.2] being real constants, [k.sub.1] > 0; [k.sub.2] > 0.

We could imagine a computer aided study of ODEs in (2.4). This may be performed by asymptotic methods or by means of numerical procedures. For our purposes, we choose the numerical way with Maple (in [4], such kind of study is made using Maple for PDEs).

Since the system in (2.4) is determined to be autonomous, we can produce direction geodesic field plots (grid of arrows tangential to solution curves) as indicated in Figures 1, 2 and 3.




For the system of first order differential equations (2.4), in Figure 4 is plotted a phase portrait of solution curves, x(t), by a numerical method. We used [k.sub.1] = 1 and the list of initial conditions x(1) [member of] {0.33; 0.66; 1.0}. Remark that the solution curve y(t) has the same shape.



In this paper we solved the problem of finding the geodesic equations [3], [18] of a class of pseudo-Riemannian manifolds, but the problem is open for other classes of manifolds. For this case study, we indicate an implicit parametric solution and use a numerical method to produce the direction geodesic field plots as well as a phase portrait of solution curves. The results our work give a relevant link between differential geometry and applied (experimental) sciences, see [1] for geometrical methods in Statistics, [2] for mathematical modeling in Ecology, [15] for optimization methods on manifolds. Regarding different but related viewpoints, the authors address the reader to these treatises and to the research works [5]/[8], [10], [11], [16] and [19] as well.

Received: April 25, 2009.


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[3] G. Bercu, C. Corcodel and M. Postolache: On a study of distinguished structures of Hessian type on pseudo-Riemannian manifolds, J. Adv. Math. Studies, 2(2009), No. 1, 1-16.

[4] Maria Teresa Calapso and C. Udriste, Isothermic surfaces as solutions of Calapso PDE, Balkan J. Geom. Appl., 13(2008), No. 1, 20-26.

[5] J. X. da Cruz Neto, O. P. Ferreira, L. R. Lucambio Perez and S. Z. Nemeth: Convex-and monotone-transformable mathematical programming problems and a proximal-like point method, JOGO, 11-32, 2003.

[6] J. Duistermaat: On Hessian Riemannian structures, Asian J. Math., 5(2001), 79-91.

[7] N. Hitchin: The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa, 25(1997), 503-515.

[8] H. Kito: On Hessian structures on the Euclidean space and the hyperbolic space, Osaka J. Math., 36(1999), 51-62.

[9] Y. Nesterov and M. J. Todd: On the Riemanian Geometry defined by self-concordant barriers and interior point methods, Found. Comp. Math., 2(2002), No. 4, 333-361.

[10] C. L. Pripoae and G. T. Pripoae: Free fall motion in an invariant field of forces, Balkan J. Geom. Appl., 14(2009), No. 1, 72-83.

[11] T. Rapcsak and T. Csendes: Nonlinear coordinate transformations for unconstrained optimization II. Theoretical Background, J. Global Opt., 3(1993), 359-375.

[12] H. Shima: Hessian manifolds of constant Hessian sectional curvature, J. Math. Soc. Japan, 47(1995), 737-753.

[13] H. Shima: The Geometry of Hessian Structures, World Scientific Publ. Co., Singapore, 2007.

[14] H. Shima and K. Yagi: Geometry of Hessian manifolds, Diff. Geom. Appl., 7(1997), 277-290.

[15] C. Udriste: Convex Functions and Optimization Methods on Riemannian Manifolds, MAIA 297, Kluwer, 1994.

[16] C. Udriste: Riemannian convexity in programming (II), Balkan J. Geom. Appl., 1(1996), No. 1, 99-109.

[17] C. Udriste and G. Bercu: Riemannian Hessian metrics, Analele Universita^ii Bucuresti, 55(2005), No. 1, 189-204.

[18] C. Udriste, G. Bercu and M. Postolache: 2D Hessian Riemannian manifolds, J. Adv. Math. Studies, 1(2008), No. 1-2, 135-142.

[19] P. M. H. Wilson: Sectional curvatures of Kahler moduli,

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Author:Bercu, Gabriel; Corcodel, Claudiu; Postolache, Mihai
Publication:Journal of Advanced Mathematical Studies
Article Type:Report
Geographic Code:4EXRO
Date:Jul 1, 2009
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