MATHEMATICAL COMMUNICATIONS 43 Math. Commun., Vol. 4, No., pp. 43-44 9) Geodesics and geodesic spheres in SL, R) geometry Blaženka Divjak,, Zlatko Erjavec, Barnabás Szabolcs and Brigitta Szilágyi Faculty of Organization and Informatics, University of Zagreb, Pavlinska, HR-4 Varaždin, Croatia Department of Geometry, Budapest University of Technology and Economics, H- 5 Budapest, Hungary Received May, 9; accepted October, 9 Abstract. In this paper geodesics and geodesic spheres in SL, R) geometry are considered. Eact solutions of ODE system that describes geodesics are obtained and discussed, geodesic spheres are determined and visualization of SL, R) geometry is given as well. AMS subject classifications: 53A35, 53C3 Key words: SL, R) geometry, geodesics, geodesic sphere. Introduction SL, R) geometry is one of the eight homogeneous Thurston 3-geometries E 3, S 3, H 3, S R, H R, SL, R), Nil, Sol. SL, R) is a universal covering group of SL, R) that is a 3-dimensional Lie group of all real matrices with determinant one. SL, R) is also a Lie group and it admits a Riemann metric invariant under right multiplication. The geometry of SL, R) arises naturally as geometry of a fibre line bundle over a hyperbolic base plane H. This is similar to Nil geometry in a sense that Nil is a nontrivial fibre line bundle over the Euclidean plane and SL, R) is a twisted bundle over H. In SL, R), we can define the infinitesimal arc length square using the method of Lie algebras. However, by means of a projective spherical model of homogeneous Riemann 3-manifolds proposed by E. Molnar, the definition can be formulated in a more straightforward way. The advantage of this approach lies in the fact that we get a unified, geometrical model of these sorts of spaces. Our aim is to calculate eplicitly the geodesic curves in SL, R) and discuss their properties. The calculation is based upon the metric tensor, calculated by E. This paper is partially supported by Croatian MSF project 6-37785-89 Corresponding author. Email addresses: blazenka.divjak@foi.hr B. Divjak), zlatko.erjavec@foi.hr Z. Erjavec), szabolcs@math.bme.hu B. Szabolcs), szilagyi@math.bme.hu B. Szilágyi) http://www.mathos.hr/mc c 9 Department of Mathematics, University of Osijek
44 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi Molnar using his projective model see [3]). It is not easy to calculate the geodesics because in the process of solving the problem we face a nonlinear system of ordinary differential equations of the second order with certain limits at the origin. We will also eplain and determine the geodesic spheres of SL, R) geometry. The paper is organized as follows. In Section we give a description of the SL, R) geometry. Further, in Section 3, the geodesics of hyperboloid model of SL, R) space are eplicitly calculated and discussed. Finally, in Section 4 the geodesic half-spheres in SL, R) are given and illustrated for radii R < π small enough.. Hyperboloid model of SL, R) geometry In this section we describe in detail the hyperboloid model of SL, R) geometry, introduced by E. Molnar in [3]. The idea is to start with the collineation group which acts on projective 3-space P 3 R) and preserves a polarity i.e. a scalar product of signature ++). Let us imagine the one-sheeted hyperboloid solid H : + + 3 3 < in the usual Euclidean coordinate simple with the origin E = ; ; ; ) and the ideal points of the aes E ; ; ; ), E ; ; ; ), E3 ; ; ; ). With an appropriate choice of a subgroup of the collineation group of H as an isometry group, the universal covering space H of our hyperboloid H will give us the so-called hyperboloid model of SL, R) geometry. We start with the one parameter group of matrices cos ϕ sin ϕ sin ϕ cos ϕ cos ϕ sin ϕ, ) sin ϕ cos ϕ which acts on P 3 R) and leaves the polarity of signature ++) and the hyperboloid solid H invariant. By a right action of this group on the point ; ; ; 3 ) we obtain its orbit cos ϕ sin ϕ; sin ϕ + cos ϕ; cos ϕ + 3 sin ϕ; sin ϕ + 3 cos ϕ), ) which is the unique line fibre) through the given point. We have pairwise skew fibre lines. Fibre ) intersects base plane E E E 3 z = ) at the point Z = + ; ; 3 ; 3 + ). 3) This action is called a fibre translation and ϕ is called a fibre coordinate see Figure ).
^R) geometry Geodesics and geodesic spheres in SL, 45 3 By usual inhomogeneous E 3 coordinates =, y =, z =, 6= fibre ) is given by µ + tan ϕ y + z tan ϕ z y tan ϕ,, y, z) 7,,,, tan ϕ tan ϕ tan ϕ where ϕ 6= π + kπ. Particularly, the fibre through the base plane point, y, z) is given by tan ϕ, y + z tan ϕ, z y tan ϕ) and through the origin by tan ϕ,, ). E 3 X,,, ) E3 E θ Z E ² ^R) Figure. Hyperboloid model of SL, The subgroup of collineations that acts transitively on the points of H and maps the origin E ; ; ; ) onto X ; ; ; 3 ) is represented by the matri 3 3 T : tji ) := 4) 3, 3 whose inverse up to a positive determinant factor Q is 3 T : tji ) = Q 3 3 3. 5) 3 Remark. µ A bijection between H and SL, R), which maps point ; ; ; ) db to matri is provided by the following coordinate transformations ca a = + 3, b = +, c = +, d = 3.
46 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi ) d b This will be an isomorphism between translations 4) and with the usual c a multiplication operations, respectively. Moreover, the request bc ad <, by using the mentioned coordinate transformations, corresponds to our hyperboloid solid + + 3 3 <. Similarly to fibre ) that we obtained by acting of group ) on the point ); ; ; 3 ) in H, a fibre in SL, cos ϕ sin ϕ R) is obtained by acting of group, on the ) sin ϕ cos ϕ d b point SL, R) see [3] also for other respects). c a Let us introduce new coordinates = cosh r cos ϕ = cosh r sin ϕ 6) = sinh r cosϑ ϕ) 3 = sinh r sinϑ ϕ)) as hyperboloid coordinates for H, where r, ϑ) are polar coordinates of the hyperbolic base plane and ϕ is just the fibre coordinate by ) and 3)). Notice that + + 3 3 = cosh r + sinh r = <. Now, we can assign an invariant infinitesimal arc length square by the standard method called pull back into the origin. Under action of 5) on the differentials d ; d ; d ; d 3 ), by using 6) we obtain the following result ds) = dr) + cosh r sinh rdϑ) + dϕ) + sinh rdϑ) ). 7) Therefore, the symmetric metric tensor field g is given by g ij = sinh rcosh r + sinh r) sinh r. 8) sinh r Remark. Note that inhomogeneous coordinates corresponding to 6), that are important for a later visualization of geodesics and geodesic spheres in E 3, are given by = = tan ϕ, y = cosϑ ϕ) = tanh r, 9) cos ϕ z = 3 sinϑ ϕ) = tanh r cos ϕ.
Geodesics and geodesic spheres in SL, R) geometry 47 3. Geodesics in SL, R) The local eistence, uniqueness and smoothness of a geodesics through any point p M with initial velocity vector v T p M follow from the classical ODE theory on a smooth Riemann manifold. Given any two points in a complete Riemann manifold, standard limiting arguments show that there is a smooth curve of minimal length between these points. Any such curve is a geodesic. Geodesics in Sol and Nil geometry are considered in [], [5] and [6]. In local coordinates u, u, u 3 ) around an arbitrary point p SL, R) one has a natural local basis {,, 3 }, where i = u. The Levi-Civita connection is i defined by i j := Γ k ij k, and the Cristoffel symbols Γ k ij are given by Γ k ij = gkm i g mj + j g im m g ij ), ) where the Einstein-Schouten inde convention is used and g ij ) denotes the inverse matri of g ij ). Let us write u = r, u = ϑ, u 3 = ϕ. Now by formula ) we obtain Cristoffel symbols Γ k ij, as follows Γ ij = cosh r) sinh r cosh r sinh r, cosh r sinh r coth r + tanh r cosh r sinh r Γ ij = Γ 3 ij = coth r + tanh r cosh r sinh r sinh r tanh r tanh r sinh r tanh r. tanh r, ) Further, geodesics are given by the well-known system of differential equations ü k + u i u j Γ k ij =. ) After having substituted coefficients of Levi-Civita connection given by ) into equation ) and by assuming first r >, we obtain the following nonlinear system of the second order ordinary differential equations r = sinhr) ϑ ϕ + sinh4r) sinhr) ) ϑ ϑ, 3) ϑ = ṙ [ ] 3 coshr) ) ϑ + ϕ, 4) sinhr) ϕ = ṙ tanh r sinh r ϑ + ϕ ). 5) By homogeneity of SL, R), we can etend the solution to limit r, due to the given assumption, as follows later on.
48 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi From 4) we get ϕ = ϑ sinhr) 4ṙ and after inserting 6) into 3) we have 3 coshr) ) ϑ, 6) r sinhr) = ϑ ϑ sinhr) coshr) ṙ ϑ ϑ. 7) Multiplying 7) by sinhr)ṙ we get a differential d ) 4ṙṙ + sinh r) ϑ ϑ = 8) dt and hence 4ṙ) + sinh r) ϑ) = 4C, 9) where C is the constant of integration, depending on initial conditions to be discussed later on. Therefore we obtain ϑ = ± C ṙ). ) sinhr) As a consequence of 3) and 4), the sign will be ) due to the geometric interpretation of a fibre translation, but we will discuss this later. From derivative of ) we get ṙ r ϑ = sinhr) ± ) C ṙ coshr) ṙ) C ṙ) sinh r). ) Further, by inserting ) and ), equation 6) has the following form r ϕ = ± ) coshr) ) ± C ṙ). ) C ṙ) sinhr) Now we put ) and ) in 5) and get ṙ r ϕ tanhr) ± ) + ± C ṙ) C ṙ) cosh r) ṙ =. 3) From this equation it follows ϕ + tanhr) ± ) C ṙ) = D, 4) where D is a new constant of integration. By equalizing ϕ from ) and 4) we have r ± ) coshr) ) ± C ṙ) C ṙ) sinhr) = D tanhr) ± ) C ṙ).
Geodesics and geodesic spheres in SL, R) geometry 49 By reordering and multiplying by ṙ sinhr) we get ṙ r ± C ṙ) sinhr) + ṙd sinhr) + ṙ coshr) ± ) C ṙ) =, which is again a differential and implies ± C ṙ) sinhr) + D coshr) = E. 5) In consistence with homogeneity we may consider lim t rt) =. This implies D = E, and relation 5) then obtains the following form ± C ṙ) = D tanh r. 6) Now from 6), ) and 4) we have respectively ṙ = ± C D tanh r, 7) ϑ = D cosh r, 8) Here we see the consistence with r ϕ = D + tanh r) = D + ϑ. 9) ṙ) = C, ϑ) = D, ϕ) = D. 3) At the same time we can assume r) =, ϑ) =, ϕ) =, as initial conditions. Further we consider the arc length t s = dτ ṙ) + cosh r) sinh r) ϑ) + ϕ + sinh r) ϑ), 3) that by 7), 8) and 9) gives s = t dτ C + D, 3) normalized with C + D = i.e. C = ṙ) = cos α, D = ϕ) = sin α and ϑ) = D = sin α can be assumed. Now, we have to consider three different cases: D = C >, D > C and C > D, with respect to the former equations as well. i) Case D = C >, or equivalently α = π 4. In this case we obtain Dt = rt) cosh ρ dρ = sinh rt), and hence rt) = arsinhdt). 33) From 8) and 9), with initial conditions ϕ) = and ϑ) =, we obtain ϑt) = arctandt), 34) ϕt) = Dt arctandt).
4 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi Particularly, C = D implies α = π 4 and hence D =. ii) Case C > D, or equivalently tan α <. From 7) we have t = rt) r) dρ rt) C D tanh ρ = cosh ρ dρ, 35) C D ) sinh ρ + C and by substitution u = C D t = sinh ρ, after integration, we obtain C D arsinh u C and hence ) C rt) = arsinh C D sinh C D t) 36) According to 8), we have ϑ = DC D DC D ) ) C cosh C D t) D = cosh C D t) C D ) + D tanh C D t), and hence by using substitution u = D tanh C D t), after integration, we get D ϑt) = arctan C D tanh C D t )). 37) Finally, from 9) we have ϕt) = D t + ϑt) and hence D ϕt) = D t arctan C D tanh C D t )). 38). z. z.4.4.6.6..5.5.5.5..4... y y.4. Figure. Geodesics in SL, R) - Case α = π 6 and α = π 4
Geodesics and geodesic spheres in SL, R) geometry 4 Figure shows geodesics through the origin for C = 3, D = and C = D = and parameter t [, ], respectively. iii) Case D > C, or equivalently tan α >. Similarly to the previous case, we start with equation t = rτ) r) dρ rτ) C D tanh ρ = cosh ρ dρ, r) C D C ) sinh ρ and by using substitution u = D C t = D C sinh ρ, after integration, we obtain arcsin u C, and hence ) C rt) = arsinh D C sin D C t). 39) From 8) we get ϑ = DD C DD C ) ) D C cos D C t) = cos D C t) D C ) + D tan D C t), and hence, by using substitution u = D tan D C t), after integration, we obtain D ϑt) = arctan D C tan D C t )). 4) Similarly to the former case ϕt) = D t + ϑt) and hence D ϕt) = D t arctan D C tan D C t )). 4) Figure 3 shows geodesic through the origin for C =, D = 3 and parameter t [, ]. Remark 3. One can easily observe special cases α =, and α = π, rs) = s, s) = ϑs) =, ys) = tanh s ϕs) =, zs) =, rs) =, s) = tan s ϑs) = s, ys) = ϕs) = s, zs) =.
4 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi y... z.4.6 Figure 3. Geodesic in SL, R) - Case α = π 3 Case D = sin α < C = cos α α < π 4 t = s H -like direction) D = C = α = π 4 t = s separating light direction) C = cos α < D = sin α π 4 < α π t = s fibre-like direction) Geodesic line hyperboloid coordinates) cos α r α s) = arsinh cos α sinh ) cos α s) sin α ϑ α s) = arctan cos α tanh ) cos α s) ϕ α s) = sin α s + ϑ α s) rs) = arsinh s ) ϑs) = arctan s ) ϕs) = s + ϑs) cos α r α s) = arsinh cos α sin ) cos α s) sin α ϑ α s) = arctan cos α tan ) cos α s) ϕ α s) = sin α s + ϑ α s) Table. Table of geodesics restricted to SL, R), s π, π ) 4. Geodesic spheres in SL, R) geometry After having investigated geodesic curves, we can consider geodesic spheres. Geodesic spheres in Sol model geometry are visualized in []. For N il geodesics, problems
^R) geometry Geodesics and geodesic spheres in SL, 43 with geodesic N il spheres and balls, and for analogous translation spheres and balls, we refer to [4], [5], [7] and [8], respectively. ^R) geometry geodesic spheres of radius R are given by following equain SL, tions XR, φ, α) = s = R, α), Y R, φ, α) = y s = R, α) cos φ z s = R, α) sin φ, ZR, φ, α) = y s = R, α) sin φ + z s = R, α) cos φ, 4) where, y, z are Euclidean coordinates of geodesics given in Table, that are transformed to formulas 9). Here φ π, π] denotes the longitude and according α π, π the altitude coordinate. For R π we consider the projective etension and the universal covering space ^R) = H by ) see [3]) for the fibre coordinate ϕ R by etra conventions. SL, That is not visual any more! In Figure 4 geodesic half-spheres in SL, R) are shown. Dark parts correspond to geodesics determined by α < π4, light parts correspond to geodesics determined by π4 < α π and black curves between these parts correspond to α = π4..6.4..4..4 z. -. -. y -.4 -.4 4 z - -4 4 3 5..5. z 4.5 -.5 -.5 y - y -. -. -4 Figure 4. Geodesic half-spheres in SL, R) of radius.5, and.5, respectively References [] A. Bo lcskei, B. Szila gyi, Visualization of curves and spheres in Sol geometry, Croat. Soc. Geo. Graph. 6), 7 3.
44 B. Divjak, Z. Erjavec, B. Szabolcs and B. Szilágyi [] A. Bölcskei, B. Szilágyi, Frenet formulas and geodesics in Sol geometry, Beiträge Algebra Geom. 487), 4 4. [3] E. Molnár, The projective interpretation of the eight 3-dimensional homogeneous geometries, Beiträge Algebra Geom. 38997), 6 88. [4] E. Molnár, On Nil geometry, Periodica Polytehnica Ser. Mech. Eng. 473), 4 49. [5] E. Molnár, J. Szirmai, Symmetries in the 8 homogeneous 3-geometries, Symmetry: Culture and Science, to appear. [6] P. Scott, The Geometries of 3-Manifolds, Bull. London Math. Soc. 5983), 4 487. [7] J. Szirmai, The densest geodesic ball packing by a type of Nil lattices, Beiträge Algebra Geom. 467), 383 397. [8] J. Szirmai, The densest translation ball packing by fundamental lattices in Sol space, Beiträge zur Algebra und Geometrie, to appear.