ON THE GEOMETRY OF CONSTANT ANGLE SURFACES IN Sol 3

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1 Kyushu J. Math. 65 (20), doi:0.2206/kyushujm ON THE GEOMETRY OF CONSTANT ANGLE SURFACES IN Sol 3 Rafael LÓPEZ and Marian Ioan MUNTEANU (Received May 200 and revised 28 April 20) Abstract. In this paper we define and classify all surfaces in the three-dimensional Lie group Sol 3 whose normals make constant angle with a left-invariant vector field.. Preliminaries The space Sol 3 is a simply connected homogeneous three-dimensional manifold whose isometry group has dimension 3 and it is one of the eight models of geometry of Thurston [4]. As a Riemannian manifold, the space Sol 3 can be represented by R 3 equipped with the metric g = e 2z dx 2 + e 2z dy 2 + dz 2, where (x, y, z) are canonical coordinates of R 3. The space Sol 3, with the group operation (x, y, z) (x,y,z ) = (x + e z x,y+ e z y,z+ z ) is a unimodular, solvable but not nilpotent Lie group and the metric g is left-invariant; see, for example, [2, 6]. This group belongs to a wider family of Lie groups, equipped with a left-invariant Riemannian metric, depending on three parameters (see, for example, [8]). Moreover, in [9], Kowalski explains the geometry of Sol 3 where it is realized as the Lie group E(, ) of rigid motions of the Minkowski plane E 2 = (R2,dxdy), endowed with the metric described above. In the Japanese literature, Sol 3 is known as Takahashi s B-manifold, see, for example, [3]. With respect to the metric g an orthonormal basis of left-invariant vector fields is given by The transformations e = e z x, e 2 = e z y, e 3 = z. () (x, y, z) (y, x, z) and (x,y,z) ( x, y, z) span a group of isometries of (Sol 3,g) having the origin as fixed point. This group is isomorphic to the dihedral group (with eight elements) D 4. It is, in fact, the complete group of isotropy [6]. The other elements of the group are (x,y,z) ( x, y, z), (x,y,z) ( y, x, z), (x,y,z) (y, x, z), (x, y, z) (y, x, z) and (x, y, z) (x, y, z).they 2000 Mathematics Subject Classification: Primary 53B25. Keywords: constant angle surfaces; homogeneous spaces. c 20 Faculty of Mathematics, Kyushu University

2 238 R. López and M. I. Munteanu can be unified as follows: (x, y, z) (±e c x + a, ±e c y + b, z + c) (x, y, z) (±e c y + a, ±e c x + b, z + c). It is well known that the isometry group of Sol 3 has dimension three. The Levi-Civita connection of Sol 3 with respect to {e,e 2,e 3 } is given by e e = e 3 e e 2 = 0 e e 3 = e e2 e = 0 e2 e 2 = e 3 e2 e 3 = e 2 e3 e = 0 e3 e 2 = 0 e3 e 3 = 0. We recall the Gauss and Weingarten formulas: (G) X Y = X Y + h(x, Y ); (W) X N = AX for every X and Y tangent to M and for any N unitary normal to M. ByA and h we denote the shape operator and the second fundamental form on M, respectively. 2. Constant angle surfaces in Sol 3 Constant angle surfaces have been studied in product spaces Q ɛ R [, 3, 4, ], where Q ɛ denotes the sphere S 2 (when ɛ =+), the Euclidean plane E 2 (when ɛ = 0), the hyperbolic plane H 2 (when ɛ = ), respectively. See also [5, 6, 0, 2, 5]. The angle θ is considered between the unit normal of the surface M and the unit tangent vector / t to the fibers of Q ɛ R, wheret is the global parameter on R. On the other hand, for Q ɛ R, theleaves {dt 0} are orthogonal to / t and they describe a foliation of the space by totally geodesic surfaces. In particular, the leaves Q ɛ {t 0 } are constant angle surfaces where the angle is θ = 0. Motivated by these facts, we consider in Sol 3 the foliations defined by totally geodesic surfaces. It is known for Sol 3 that there are only two such foliations, namely, H ={dx 0} and H 2 ={dy 0}. Each leaf of these foliations is isometric to the hyperbolic plane. Let us consider H. It follows that the tangent plane to H (the leaf at each x = x 0 )is spanned by / y and / z, while the unit normal vector is e. Similarly, for the foliation H 2, the vector e 2 is the unit normal vector to each leaf y = y 0. Due to these reasons, we give the following definition. Definition 2.. An oriented surface M, isometrically immersed in Sol 3, is called a constant angle surface with respect to H (respectively H 2 ) if the angle between its unit normal vector and e (respectively e 2 ) is constant in each point of the surface M. As it occurs in Q ɛ R, the leaves of the foliation of Sol 3 are constant angle surfaces. Exactly, the surfaces x = x 0 (respectively y = y 0 ) are constant angle surfaces with respect to H with θ = 0 (respectively θ = π/2) and constant angle surfaces with respect to H 2 with θ = π/2 (respectively θ = 0). In this article we classify all constant angle surfaces in Sol 3. Because we have two types of constant angle surfaces (with respect to H or with respect to H 2 ), from now we are only going to consider constant angle surfaces with respect to H and from now, we abbreviate by

3 Constant angle surfaces in Sol saying a constant angle surface: the computations of constant angle surfaces with respect to H 2 are similar and since the differences are insignificant we do not give any details for this problem. Remark 2.2. The vector field e 3 = / z is, together with e and e 2, the third left-invariant Killing vector field of Sol 3 (see ()). Although the vector field e 3 defines in Sol 3 a foliation, its leaves are not totally geodesic surfaces but minimal surfaces. Following the motivation of our definition of constant angle surface from what happens in Q ɛ R, we discard in our study any consideration with the vector field e 3. Let M be a constant angle surface in Sol 3. Denote by θ [0,π)the angle between the unit normal N and e. Hence g(n, e ) = cos θ. Let T be the projection of e on the tangent plane T p M of M in a point p M. Thus e = T + cos θn. (2) 3. First computations and some particular cases First we consider the case θ = 0. Then N = e and hence the surface M is isometric to the hyperbolic plane H ={dx 0}. PROPOSITION 3.. The only constant angle surfaces in Sol 3 with θ = 0 are the leaves of the foliation H, that is, the surfaces given by x = x 0, x 0 R. We now suppose that θ 0. LEMMA 3.2. If X is tangent to M we have: (i) X e = g(x, e )e 3, X e 2 = g(x, e 2 )e 3, X e 3 = g(x, e )e g(x, e 2 )e 2 ; (ii) AT = g(n, e 3 )T, hence T is a principal direction on the surface; (iii) g(t, T ) = sin 2 θ. At this point we also have to decompose e 2 and e 3 into the tangent and the normal parts, respectively. As θ 0, let E = (/sin θ)t. Consider E 2 tangent to M, orthogonal to E and such that the basis {e,e 2,e 3 } and {E,E 2,N} have the same orientation. It follows that e = sin θe + cos θn e 2 = cos α cos θe + sin αe 2 cos α sin θn (3) e 3 = sin α cos θe + cos αe 2 + sin α sin θn and E = sin θe + cos θ cos αe 2 cos θ sin αe 3 E 2 = sin αe 2 + cos αe 3 (4) N = cos θe sin θ cos αe 2 + sin θ sin αe 3 where α is a smooth function on M.

4 240 R. López and M. I. Munteanu We now study constant angle surfaces in the case that θ = π/2. In this case, e is tangent to M and T = E. The metric connection on M is given by E E = cos αe 2, E2 E = 0 E E 2 = cos αe E2 E 2 = 0. The second fundamental form is obtained from h(e,e ) = sin αn, h(e,e 2 ) = 0, h(e 2,E 2 ) = σn, where σ is a smooth function on M. Writing the Gauss formula (G) for X = E and Y = E 2, respectively for X = Y = E 2, one obtains E (α) = 0 and E 2 (α) = sin α σ. Remark 3.3. The surface M is minimal if and only if σ = sin α.sincee and E 2 are linearly independent, it follows that α is constant. Moreover, M is totally geodesic if and only if α = 0, in which case M coincides with H. Due to the fact that the Lie bracket of E and E 2 is [E,E 2 ]=cos αe, one can choose local coordinates u and v such that E 2 = and E = β(u, v) u v. This choice implies α and β fulfill the partial differential equation (PDE) β u = β cos α. Since α depends only on u, it follows that where ρ is a smooth function depending on v. Denote by β(u, v) = ρ(v)e u cos α(τ) dτ, F : U R 2 M Sol 3 (u, v) (F (u, v), F 2 (u, v), F 3 (u, v)) the immersion of the surface M in Sol 3.Wehave: (i) u = F u = (F,u,F 2,u,F 3,u ) = E 2 = (sin αe 2 + cos αe 3 ) F(u,v) = (0,e F3(u,v) sin α, cos α); (ii) v = F v = (F,v,F 2,v,F 3,v ) = β E = ( ) β e F(u,v) = β e F3(u,v), 0, 0.

5 Constant angle surfaces in Sol 3 24 It follows that Thus we obtain F = F (v), v F = u F 2 = sin α(u)e F3(u,v), u F 3 = cos α(u), v F (v) = ρ(τ) dτ, F 2 = F 2 (u), F 3 = F 3 (u). β(u, v) e F 3(u,v), F 2 (u) = u (sin α(τ)e τ cos α(s) ds )dτ, F 3 (u) = u cos α(τ) dτ. Changing the v parameter, one gets the following. PROPOSITION 3.4. Let M be a constant angle surface in Sol 3 with θ = π/2. Then it may be parametrized as F(u,v)= (v, φ(u), χ(u)) which represents a cylinder over the planar curve γ(u)= (0, φ(u), χ(u)), where φ(u) = u (sin α(τ)e τ cos α(s) ds )dτ and χ(u) = u cos α(τ) dτ. Moreover, the surface is the group product between the curve v (v, 0, 0) and the curve γ. (a) InFigurewecanseehowthecurveγ looks for different values of the function α: α is a constant: γ(u)= (0, tan αe u cos α,ucos α); (b) α(s) = s: (c) α(s) = s 2 : ( u ) γ(u)= 0, sin se sin s ds, sin u ; ( u γ(u)= 0, sin s 2 s u ) cos τ e 2 dτ ds, cos s 2 ds ; (d) α(s) = arccos(s), s [, ]: γ(u)= ( u ) 0, s 2 e s ds, u ; (e) α(s) = 2arctane 2s : In this case, the expression of γ involve hypergeometric functions. The surface M is totally umbilical but not totally geodesic.

6 242 R. López and M. I. Munteanu (b) (c) (d).5 (e) FIGURE. The curve γ according to items (b), (c), (d) and (e). Returning to the general case for θ, we distinguish some particular situations for α. Assume that sin α = 0. Then cos α =± and the principal curvature corresponding to the principal direction T vanishes. Straightforward computations yield θ = π/2, which was discussed before. We suppose that cos α = 0. Then sin α =± and the relations (2) and (4) may be written in an easier way. Assume sin α = (similar results in the case sin α = ). We have e = sin θe + cos θn, e 2 = E 2, e 3 = cos θe + sin θn E = sin θe cos θe 3, E 2 = e 2, N = cos θe + sin θe 3. The Levi-Civita connection on the surface M is given by E E = 0, E E 2 = 0, E2 E = cos θe 2, E2 E 2 = cos θe. Such a surface is minimal, because the second fundamental form is h(e,e ) = sin θn, h(e,e 2 ) = 0, h(e 2,E 2 ) = sin θn and hence its mean curvature vanishes.

7 Constant angle surfaces in Sol In order to obtain explicit embedding equations for the surface M let us choose local coordinates as follows. Let u be such that E = / u and v such that E 2 and / v are collinear. This can be done due to the fact that [E,E 2 ]= cos θe 2. Considering / v = b(u, v)e 2, with b a smooth function on M,since[ / u, / v]=0, it follows that b satisfies b u b cos θ = 0. This PDE has the general solution b(u, v) = μ(v)e u cos θ, with μ a smooth function defined on a certain interval in R. Denote by F = (F,F 2,F 3 ) the isometric immersion of the surface M in Sol 3.Wehave: (i) u = F u = ( u F, u F 2, u F 3 ) = E = sin θe F(u,v) cos θe 3 F(u,v) = (sin θe F3(u,v), 0, cos θ); (ii) v = F v = ( v F, v F 2, v F 3 ) = μ(v)e u cos θ E 2 = μ(v)e u cos θ e 2 F(u,v) = (0, μ(v)e u cos θ+f3(u,v), 0). Looking at (i) we immediately get: the third component: F 3 (u, v) = ucos θ + ζ(v), ζ C (M); the second component: F 2 (u, v) = F 2 (v). Replacing in (ii) we obtain: the third component: ζ(v) = ζ 0 R; the second component: F 2 (v) = e ζ 0 v μ(τ) dτ; the first component: F (u, v) = F (u). Going back to (i) and considering the first component one gets F (u) = e ζ 0 tan θe u cos θ + constant. Since the map (x, y, z) (x + c, y, z) is an isometry for Sol 3, we can take the previous constant to be 0. Moreover, the map (x, y, z) (e c x, e c y, z + c) is also an isometry of the ambient space, so ζ 0 mayalsobeassumedtobe0. Consequently, one obtains the following parametrization for the surface M: ( v ) F(u,v)= tan θe u cos θ, μ(τ) dτ, u cos θ. Finally, we may change the parameter v such that μ(v) =. One can state the following. PROPOSITION 3.5. Consider M a constant angle surface in Sol 3 with angle θ and θ 0,π/2.Ifcos α = 0 in (3) and (4), then parametrization of M is F(u,v)= (tan θe u cos θ,v, ucos θ). (5) Moreover this surface is minimal and it is the group product between the curve v (0,v,0) and the planar curve γ(u)= (tan θe u cos θ, 0, u cos θ).

8 244 R. López and M. I. Munteanu 4. The general classification theorem From now on we will deal with θ 0,π/2andcosα 0, sin α 0. LEMMA 4.. The Levi-Civita connection on M and the second fundamental form h are given by { E E = cos αe 2, E E 2 = cos αe, (6) E2 E = σ cot θe 2, E2 E 2 = σ cot θe, h(e,e ) = sin θ sin αn, h(e,e 2 ) = 0, h(e 2,E 2 ) = σn. (7) The matrix of the Weingarten operator A with respect to the basis {E,E 2 } has the expression ( ) sin α sin θ 0 A = 0 σ for a certain function σ C (M). Moreover, the Gauss formula yields and the compatibility condition E (α) = 2cosθ cos α, E 2 (α) = sin α σ sin θ, (8a) (8b) ( E E 2 E2 E )(α) =[E,E 2 ](α) = E (E 2 (α)) E 2 (E (α)) gives rise to the differential equation E (σ ) + σ cos θ sin α + σ 2 cot θ = 2sinθ cos θ sin 2 α. (9) We are looking for a coordinate system (u, v) in order to determine the embedding equations of the surface. Let us take the coordinate u such that / u = E. We will consider v later. Let us turn our attention to (8a) which can be re-written as Solving this PDE one gets u α = 2cosθ cos α. sin α = tanh(2u cos θ + ψ(v)), where ψ is a smoothfunctionon M dependingon v. Notice that, apparently, the equation also has a second solution sin α = coth(2u cos θ + ψ(v)). This is not valid because coth takes values in (, ) or in (, + ). Now, let us take v in such way that α/ v = 0, namely ψ is a constant; denote it by ψ 0. It follows that α is given by sin α = tanh(ū), (0) where ū = 2u cos θ + ψ 0. At this point, equation (9) becomes σ u + cot θ(σ + 2sinα sin θ)(σ sin α sin θ) = 0. ()

9 Constant angle surfaces in Sol Since / v is tangent to M, it can be decomposed in the basis {E,E 2 }. Thus, there exist functions a = a(u, v) and b = b(u, v) such that v = ae + be 2. Due to the choice of the coordinate v we have 0 = α ( = 2a cos θ cos α + b sin α σ ). v sin θ If b = 0, then cos θ = 0orcosα = 0 and both situations were already studied. Consider now b 0. Let us denote p(u, v) = a/b. Hence the equality above yields σ = sin θ sin α + p sin 2θ cos α. (2) On the other hand [ 0 = u, ] = a u E + b u E 2 + b(cos αe σ cot θe 2 ). v Subsequently { au + b cos α = 0, (3) b u bσ cot θ = 0. If in (2) we take the derivative with respect to u, and combining with (), it follows that p u + cos α + p cos θ sin α + 2p 2 cos 2 θ cos α = 0. (4) Straightforward computations yield the general solution for this equation (see the appendix), namely p(u, v) =±, (5) cosh 3 2 ū cos θ sinh ū + ε I(u)+ (v) where ε = 0, and is a certain function depending on v. Let F : U R 2 M Sol 3, (u, v) (F (u, v), F 2 (u, v), F 3 (u, v)) be the immersion of the surface M in Sol 3.Wehave I. u = F u = (F,u,F 2,u,F 3,u ) = E = sin θe F(u,v) + cos θe 2 F(u,v) cos θ sin αe 3 F(u,v) which implies = (sin θe F 3(u,v), cos θ cos αe F 3(u,v), cos θ sin α), u F = sin θe F 3(u,v), u F 2 = cos θ cos αe F3(u,v), u F 3 = cos θ sin α. (6a) (6b) (6c) From the last equation one immediately obtains F 3 (u, v) = 2 log cosh(ū) + ζ(v), (7)

10 246 R. López and M. I. Munteanu where ζ is a smooth function. Replacing this expression in (6a) and (6b), one gets where I(u)= u cosh(2τ cos θ + ψ0 )dτ, J(u)= F = sin θe ζ(v) (I (u) + f (v)), (8) F 2 =±cos θe ζ(v) (J (u) + f 2 (v)), (9) u cosh 3 2 (2τ cos θ + ψ0 )dτ (20) and f, f 2 are some smooth functions which will be determined in what follows. II. v = F v = (F,v,F 2,v,F 3,v ) = a(u, v)e + b(u, v)e 2 = a(u, v)(sin θe F(u,v) + cos θ cos αe 2 F(u,v) cos θ sin αe 3 F(u,v) ) +b(u, v)(sin αe 2 F(u,v) + cos αe 3 F(u,v) ). It follows that v F = a(u, v) sin θe F 3(u,v), v F 2 = (a(u, v) cos θ cos α + b(u, v) sin α)e F3(u,v), v F 3 = a(u, v) cos θ sin α + b(u, v) cos α. (2a) (2b) (2c) From (7) and (2c) we have and from (8) and (2a) we obtain a(u, v) cos θ sin α + b(u, v) cos α = ζ (v) ζ (v)(i (u) + f (v)) f (v) + a(u, v) cosh(ū) = 0. (22) Taking the derivative with respect to u, one gets Equation (23) in a has the solution ζ (v) + a u (u, v) + a(u, v) cos θ tanh(ū) = 0. (23) a(u, v) = ζ (v)i (u) + ξ(v) cosh(ū) (24) yielding ( ) cos θ sinh(ū) b(u, v) =± ( ζ (v)i (u) + ξ(v)) + ζ (v) cosh(ū). (25) cosh(ū) Recall that p(u, v) = a(u, v)/b(u, v). We immediately notice that the general solution given by (5) is obtained with the following identification: ε = 0 ζ (v) = 0andε = (v) = ξ(v)/ζ (v). It follows that p(u, v) =± cos θ sinh(ū) + ζ (v) cosh 3 2 ū ζ (v)i (u)+ξ(v).

11 Constant angle surfaces in Sol At this point we will obtain the parametrization of the surface in the following way.. Combining (24) with (22) one gets f (v) ζ (v)f (v) ξ(v) = 0 which has the solution v f (v) = e ζ(v) ξ(τ)e ζ(τ) dτ. Thus ( v ) F (u, v) = sin θ e ζ(v) I(u)+ ξ(τ)e ζ(τ) dτ. 2. Similarly, replace (9) in (2b) and one obtains We have cos θ(f 2 (v) + ζ (v)f 2 (v)) ( + ζ (v) cos θ(i(u) + J(u)) sinh(ū) ) = cos θ ξ(v). (26) cosh(ū) a(u, v) cos θ cos α + b(u, v) sin α =±ζ (v)(sinh(ū) cos θi (u) cosh(ū)) ± cos θ cosh ūξ(v) and cos θ(i(u) + J(u)) sinh(ū) cosh(ū) = constant which can be incorporated in the primitives I(u) or J(u). It follows that f 2 satisfies the following ordinary differential equation (ODE) f 2 (v) + ζ (v)f 2 (v) = ξ(v) which has the solution v f 2 (v) = e ζ(v) ξ(τ)e ζ(τ) dτ. Thus ( v ) F 2 (u, v) =±cos θ e ζ(v) J(u)+ ξ(τ)e ζ(τ) dτ. We conclude with the following result that classifies the constant angle surfaces in the general case. THEOREM 4.2. A general constant angle surface in Sol 3, with θ 0,π/2 and cos α 0, can be parameterized as F(u,v)= γ (v) γ 2 (u), (27) where ( v v γ (v) = sin θ ξ(τ)e ζ(τ) dτ, ± cos θ ξ(τ)e ζ(τ) dτ, ζ(v)), (28a) γ 2 (u) = (sin θi (u), ± cos θj (u), 2 log cosh ū), (28b) where I and J are defined in (20). Here ζ, ξ are arbitrary functions depending on v and ū = 2u cos θ + constant. COROLLARY 4.3. The only minimal constant angle surfaces in Sol 3 are the hyperbolic planes x = x 0 (for θ = 0), the hyperbolic planes y = y 0 (for θ = π/2) and the surfaces given in Proposition 3.5.

12 248 R. López and M. I. Munteanu Proof. In the general case when θ is different from 0 and π/2 andα is such that sin α and cos α do not vanish, the minimality condition can be written as σ = sin α sin θ. But this relation is impossible due to (2) and (4). Appendix. Solution of the PDE We prove that the solution of p u + cos α + p cos θ sin α + 2p 2 cos 2 θ cos α = 0 is given by (5), where θ 0,π/2andcosα 0. Denote by ū = 2u cos θ + ψ 0. Let q := /p; it follows that q satisfies q u q 2 cos α q cos θ sin α 2cos 2 θ cos α = 0. Let A := q cos θ sinh ū. It follows that q u = A u + 2cos 2 θ cosh ū. Hence, A satisfies A u 3A cos θ tanh ū cosh ū A2 = 0. Let B := A cosh 3 2 ū. It follows that A u = 3B cos θ sinh ū cosh 2 ū + B u cosh 3 2 ū. Thus, B satisfies B u B 2 cosh 2 ū = 0. Hence either B = 0or/B(u, v) = I(u)+ (v), for a smooth. () If B = 0thenA = 0, q = cos θ sinh ū. Thenq 0 if and only if θ π/2 andū 0. One gets p = cos θ sinh ū, obtaining (5). (2) If B 0then cosh 3 2 ū q(u, v) = cos θ sinh ū + I(u)+ (v). These solutions correspond to. ζ = 0and2. (v) = ξ(v)/ζ (v). Acknowledgements. The authors wish to thank the anonymous referee for some useful comments which helped us to improve the quality of the paper. The first author was partially supported by MEC-FEDER grant no. MTM and Junta de Andalucía grant no. P06-FQM The second author was partially supported by Grant PN II ID 398/ (Romania). REFERENCES [] P. Cermelli and A. J. Di Scala. Constant-angle surfaces in liquid crystals. Philosophical Magazine 87 (2007), [2] B. Daniel and P. Mira. Existence and uniqueness of constant mean curvature spheres in Sol 3. Preprint, arxiv: v2. [3] F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken. Constant angle surfaces in S 2 R. Monatsh. Math. 52 (2007),

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