SLANT AND LEGENDRE CURVES IN BIANCHI-CARTAN-VRANCEANU GEOMETRY

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1 KOREKTURY cmj-4473.tex SLANT AND LEGENDRE CURVES IN BIANCHI-CARTAN-VRANCEANU GEOMETRY Constantin Călin, Mircea Crasmareanu, Iaşi Received July 3, 3 Abstract. We study Legendre and slant curves for Bianchi-Cartan-Vranceanu metrics. These curves are characterized through the scalar product between the normal at the curve and the vertical vector field and in the helix case they have a proper non-harmonic mean curvature vector field. The general expression of the curvature and torsion of these curves and the associated Lancret invariant for the slant case are computed as well as the corresponding variant for some particular cases. The slant particularly Legendre curves which are helices are completely determined. Keywords: Bianchi-Cartan-Vranceanu metric; slant curve; Legendre curve; Lancret invariant; helix MSC : 53D5, 53B5, 53A55, 53C5. Preliminaries Among the Riemannian manifolds of non-constant sectional curvature a special rôle is played by the homogeneous spaces with a large isometry group. Due to the recent approach of Hamilton-Perelman to the Poincaré conjecture by means of Ricci flow, great interest is paid to dimension three. It is well-known that the maximum dimension of the isotropy group of a 3- dimensional manifold is 6 and that there is no metric with 5-dimensional group. The Bianchi-Cartan-Vranceanu spaces are certain 3-dimensional homogeneous Riemannian manifolds with 6- and 4-dimensional isometry group. They form a two parameters family with parameters denoted as l and m containing, among others, some remarkable 3-manifolds: R 3, S 3, S R, H R and the 3-dimensional Heisenberg group Nil 3. Recently, several studies have been devoted to special submanifolds in these spaces: parallel surfaces [3], biharmonic curves [7], [3] and [4], constant

2 angle surfaces [6], graphs of constant mean curvature [], biharmonic surfaces [3], higher order parallel and totally umbilical surfaces [5]. The aim of this paper is to continue the study of submanifolds, more precisely -dimensional submanifolds. In fact, two types of curves are discussed in these spaces: θ-slant curves with θ an arbitrary angle considered between the tangent vector field and the vertical vector field E 3 = / z and particularly at θ = π/, Legendre curves. We choose this subject for two reasons: the rôle of Legendre curves in almost contact geometry is remarkable and wellknown; in [5] the reader finds an excellent survey on these curves, although the literature on Legendre curves is rich [4], [6], [8], [7], [], [6], [7], slant curves have been studied until now only for the Sasakian geometry in [3], for the contact pseudo-hermitian geometry in [5], for the f-kenmotsu geometry in [], in normal almost contact geometry in [9] and for warped products in []. Although some Bianchi-Cartan-Vranceanu metrics are almost contact metrics we prefer in the present work a unified treatment in order to emphasize the common properties of these metrics. Another feature of the present paper, as compared with [9], is that here we do not use the structural tensor field ϕ of, -type, which is a main tool in almost contact geometry. In Section 3 we obtain a characterization for all these curves in terms of orthogonality between the E 3 and the normal vector field. Based on this result we derive the expression of the Frenet frame as well as the curvature and torsion. By defining the Lancret invariant of a θ-slant curve γ with θ, π as. Lancretγ = cos θ sin θ, we obtain the expression of this invariant in the Bianchi-Cartan-Vranceanu setting. Also, in Section 3 we prove an important property of slant non-legendre curves: in the non-geodesic case and for l these are Bertrand curves, i.e., there exists an affine dependence between curvature and torsion. In the case m = we provide the general expression of slant particularly Legendre curves and their curvature, torsion and Lancret invariant, as well as of slant helices. The last section is devoted to examples and we completely discuss three manifolds: the Euclidean space, the Heisenberg group Nil 3 and the 3-dimensional sphere S 3. Some results already known from [9] and [3] are reobtained with particular choices of the parameters; but to the best of our knowledge, until now there have been no general expressions for Legendre curves in Nil 3 nor for their curvature and torsion in S 3.

3 . Bianchi-Cartan-Vranceanu 3 geometries and curves in Riemannian setting Fix two real numbers l and m and denote by M 3 m the manifold {x, y, z R 3 ; F x, y, z = + mx + y > }. We shall consider on M 3 m the Bianchi-Cartan- Vranceanu metric [5], page 343,. g l,m = F dx + F dy + dz + ly lx. dx F F dy An important feature of these metrics is their S -invariance, i.e., the invariance with respect to transformations. x cos ϕ sin ϕ x ỹ = sin ϕ cos ϕ y. z z Other remarks concerning these metrics are in [5] and the particular cases are discussed here in the last section. An orthonormal basis is.3 E = F x ly z, E = F y + lx z, E 3 = z with the dual basis.4 ω = dx F, ω = dy F, ω 3 = dz + ly lx dx F F dy. We need the covariant derivatives of E i ; if is the Levi-Civita connection of g l,m we have.5.6 E E = mye, E E = mxe, E3 E 3 = { E E = mye + le 3, E E = mxe le 3 E E 3 = E3 E = le, E E 3 = E3 E = le. From.5 3 it results that E 3 is a geodesic vector field, i.e., its integral curves are geodesics of g l,m. In the following we call this vector field vertical. A straightforward computation yields that E 3 is also a Killing vector field; for the set of all Killing vector fields see [4]. Next, following [5], page 64, we recall the notion of a Frenet curve in a n + - dimensional manifold: Let m be an integer with m n +. The curve γ : I R M parametrized by the arc length s is called an m-frenet curve on M 3

4 if there exist m orthonormal vector fields U = γ, U,..., U m along γ and m positive smooth functions k,..., k m of s such that:.7 γ U = k U, γ U = k U + k U 3,..., γ U m = k m U m. The function k j is called the j-th curvature of γ, and γ is a a geodesic if m = ; then we get the well-known equation γ γ = ; b a circle if m = and k is a constant: then we have γ E = k E, γ E = k E ; c a helix of order m if k,..., k m are constants. The Frenet curve γ is called non-geodesic if k > everywhere on I and in dimension 3 it is called a generalized helix if k /k = const. 3. Slant and Legendre curves in Bianchi-Cartan-Vranceanu geometry Let γ be a 3-Frenet curve in M 3 m, g l,m for which we denote the Frenet frame as usual T = γ, N, B and let us consider the Frenet equations 3. a T T = kn, b T N = kt + τb, c T B = τn, where k denotes the curvature of the curve γ and τ is its torsion. The main notion of this paper is introduced as follows: Definition. The vertical angle of γ is the function θ : I [, π given by: 3. cos θs = g l,m T s, = g l,m T s, E 3. z The curve γ is called a slant curve or more precisely θ-slant curve if θ is a constant function. In the particular case of θ = π/ the curve γ is called a Legendre curve. In the following we suppose that γ is non-geodesic, i.e., k > and then γ cannot be an integral curve of E 3 which means θ, π. Let γ s = γ s, γ s, γ 3s be the general form of T s whose expression in the given frame [ 3.3 T = γ F E + γ F E lγ γ γ γ ] + + γ 3 E 3. F Now we are able to prove the first important result: 4

5 Theorem 3.. A non-geodesic curve γ in M 3, g l,m is a θ-slant curve if and only if its normal vector field N is g l,m -orthogonal to E 3 : 3.4 ω 3 N =. This yields the decomposition of E 3 in the Frenet frame: 3.5 E 3 = cos θt + sin θ B. Thus, a Legendre curve has B = E 3. P r o o f. From 3.3 and.6 we have 3.6 γ E 3 = lγ F E lγ F E. We derive covariantly the relation 3.: 3.7 = θ sin θ = g l,m kn, E 3 + g l,m T, lγ E lγ E and then kω 3 N =. An important consequence of 3.5 is the expression of the Frenet frame with respect to E i : 3.8 T as in 3.3, N = ± F sin θ γ E + γ E, B = cos θ F sin θ γ E + γ E + sin θ E 3 where for obtaining the above formulae we use two other equations: the θ-slant condition reads 3.9 lγ γ γ γ F the unit length of γ yields via γ 3 = cos θ 3. γ + γ = F sin θ. The sign of N is fixed by the positivity of k and τ. In [9], page 55, the following notion is introduced: a non-geodesic curve is called a slant helix if the principal normal lines of γ make a constant angle with a fixed direction. Therefore, a slant curve is a slant helix with E 3 as the fixed direction. The second main result gives the curvature and torsion of γ: 5

6 Theorem 3.. If γ : I M 3, g l,m is a θ-slant non-geodesic curve then its curvature and torsion are k = γ γ γ γ lγ 3 sin θ + γ γ γ γ 4m l sin θ, F sin θ F τ = l + γ γ γ γ F sin lγ 3 θ F + γ γ γ γ 4m l cos θ. F It follows that the Lancret invariant of slant curves in Bianchi-Cartan-Vranceanu geometry is 3.3 Lancret ± γ = τ ± l k. A Legendre curve has 3.4 k = F γ γ γ γ F P r o o f. We prove first that 3.5 k = δ sin θ, τ = where 3.6 δ = From the first Frenet equation we have lγ 3 + γ γ γ γ 4m l l + δ cos θ, F sin θ g l,m γ γ, γ E + γ E. 3.7 k = g l,m γ γ, N and the expression 3.8 of N yields 3.5. By 3.5 we have 3.8 B = sin θ E 3 cos θ sin θ T and then, τ = l. 3.9 γ B = sin θ γ E 3 cos θ sin θ γ T = l F sin θ [γ E γ E ] cos θ sin θ kn, which gives 3.5 by choosing + in 3.8. We deduce that cos θ/ sin θ = τ ± l//k, which gives the Lancret invariant. 6

7 A long but straightforward computation yields 3. δ = γ γ γ γ F sin θ F lγ 3 sin θ + γ γ γ γ 4m l sin θ since 3. g l,m γ γ, γ E = d γ γ ds F g l,m γ γ, γ E = d γ γ ds F + γ γ F + γ g l,m γ, γ E, γ γ F γ g l,m γ, γ E. We have 3. [ 8m l γ E γ γ γ γ = 4F [ l γ E 8mγ γ γ γ = 4F and hence we get the conclusion. ] lγ 3 E lγ + lγ 3 F E 3 ] E + lγ F E 3. Corollary. Suppose that l. A θ-slant curve in M 3, g l,m which is nongeodesic and non-legendre is a Bertrand curve. P r o o f. Recall that γ is a Bertrand curve if there exists a, b R \ {} such that 3.3 ak + bτ =. From 3.5 we get δ = k sin θ, ±τ = l + δ cos θ. Expressing δ from both the relations we have k cos θ/sin θ + ±τ = l/, which yields 3.3 with a = cos θ/l sin θ and b = ±/l. The case when the equations are explicitly integrable is m = : Theorem 3.3. Let γ be a non-geodesic θ-slant curve in R 3, g l,. Then there exists a unit-length parametrization ζs = cos us, sin us of the unit circle S such that γ = γ u has the expression s 3.4 γ u s = sin θ ζt dt, s cos θ l [ sin θ s cos ut t ] sin ut cos uϱ dϱ dt. t sin uϱ dϱ 7

8 Its curvature, torsion and Lancret invariant are 3.5 ks = u s l cos θ sin θ l sin θ γ sγ s γ sγ s, τs = l u + s l cos θ l γ sγ s γ sγ s cos θ, Lancretγ = τ ± l k. In particular, a Legendre curve in R 3, g l, has the expression s 3.6 γu L s = ζt dt, l [ s cos ut and its curvature and torsion are t sin uϱ dϱ sin ut 3.7 ks = u s l γ sγ s γ sγ s, τs = l. t ] cos uϱ dϱ dt, P r o o f. Due to F = the relation 3. implies the existence of ζ = ζs such that γ = sin θ cos us, γ = sin θ sin us and the integration gives 3.4. A straightforward computation yields the conclusion. The particular case us = ps with p gives the corresponding θ-slant curve sin θ 3.8 γ p s = sinps, cosps, s cos θ l sin θ sinps s p p p with 3.9 ks = p l cos θ sin θ l sin θ 3 cosps, p τs = l [ + p l cos θ l sin θ ] cosps cos θ p and the Legendre curve 3.3 γ L p s = p 8 sinps, cosps, l sinps s p

9 with 3.3 ks = p l cosps, τs = l p. Denote by h the second fundamental form of γ and by H its mean curvature field 3.3 H = traceh = ht, T = T T. Then γ is called a curve with proper mean curvature vector field if there exists λ C γ such that 3.33 H = λh. In particular, if λ = then γ is known as a curve with harmonic mean curvature vector field. Here the Laplace operator acts on the vector valued function H and is given by 3.34 H = T T T T. Making use of Frenet equations, we can rewrite 3.33 as k kt + k k 3 kτ N + k τ + kτ B = λkn. It follows that both k and τ are constants, and the function λ becomes a constant too, namely 3.36 λ = κ + τ see also Theorem. in [8]. In our framework we state the following result: Theorem 3.4. A non-geodesic θ-slant curve γ in M 3, g l,m has a proper mean curvature vector field if and only if γ is a helix and then 3.37 λ := λ θ = δ + l 4 + l δ cos θ. In particular, a helix Legendre curve satisfies 3.38 λ := λ Legendre = δ + l 4 >. P r o o f. We compute λ of 3.36 by using

10 Since δ appears as a main expression in all computations above we are interested in its other form. So, inspired by [7], we introduce the angle function β = βs given by 3.39 γ = F sin θ cos β, γ = F sin θ sin β. Then one obtains 3.4 δ = β + m sin θγ cos β γ sin β l cos θ. Using this expression we can characterize the θ-slant helices in Bianchi-Cartan- Vranceanu manifolds. Namely, k and τ are constants if and only if δ is a constant and this condition δ = reads 3.4 β β = F F and then aβ s = F γs with a a positive number. Then we integrate 3.39: Theorem 3.5. The θ-slant helices in M 3 m, g l,m are given by 3.4 γ a s = a sin θ sin βs + c, a sin θ cos βs + c, s cos θ + l sin θ s s ] [as sin θ + c sin βt dt c cos βt dt where c, c are real constants; here, β = βs is a solution of the ordinary differential equation 3.43 aβ = + m[a sin θ sin β + c + a sin θ cos β + c ] which determines also the possible F s where 3.4 holds. For c = c = we get 3.44 γ a s = a sin θ sin a + ma sin θ s, a sin θ cos a + ma sin θ s, with 3.45 k = a ma sin θ l cos θ sin θ, τ = l + a ma sin θ l cos θ cos θ. cos θ + la sin θ s

11 The Legendre helices are 3.46 γ a s = with a sin a + ma s, a cos a + ma s, la s 3.47 k = a ma, l τ =, λ Legendre = a ma l Examples 4.. Euclidean geometry. E 3 : m = l = become 4. γ u s = with sin θ s ζt dt, s cos θ 4. ks = u s sin θ = γ γ γ γ, τs = u s cos θ, Lancretγ = τ sin θ k become s 4.3 γu L s = with ζt dt, 4.4 ks = u s = γ γ γ γ, τs =. It is a well known fact that the curvature of a unit-speed plane curve is ks = γ γ γ γ. We get that γ u from 4. is a helix if and only if u is a constant, say p. Then γ u is exactly γ p given by 3.8 with l = and then λ θ = λ Legendre = λ u = u = p. Another equivalent expression is given by 3.44: 4. helices γ a s = a sin θ sin s a, a sin θ cos s a, s cos θ with k = sin θ /a and τ = cos θ /a.

12 4.. The Heisenberg group. Nil 3 : m =, l = become 4.5 γ u s = sin θ s ζt dt, s cos θ [ s + sin θ cos ut t sin uϱ dϱ sin ut t ] cos uϱ dϱ dt with 4.6 ks = u s + cos θ γ sγ s + γ sγ s sin θ, τs = u s + cos θ γ sγ s + γ sγ s cos θ, Lancretγ = τ ± /k. The above Lancret invariant was obtained for the general Sasakian 3-dimensional geometry in [3], page become s 4.7 γu L s = ζt dt, with s cos ut t sin uϱ dϱ sin ut 4.8 ks = u s γ sγ s γ sγ s, τs = read 4.9 λ θ = δ + δ cos θ, λ Legendre = δ + where, from 3.4, δs = u s + cos θ. The θ-slant helices 3.44 are 4.9 helices γ a s = t cos uϱ dϱ dt a sin θ sin s a, a sin θ cos s a, cos θ a sin θs with k = sinθ /a and τ = /a cos θ cos θ while the Legendre helices are 4. helices γ a s = a sin s a, a cos s a, as. The proper biharmonic curve of [3], page 364, is a γ u of 4.5 with us = As + a while the non-helix slant curve of [3], page 365, corresponds to us = ln s.

13 4.3. The sphere. For 4m = l we have S 3 m. In particular, m =, l = gives S become for S while 3.3 gives γ ks = γ γ γ sin θ + γ + γ + γ + + γ γ 3 sin θ, τs = γ γ γ γ sin θ + γ + γ + γ 3 + γ + cos θ γ 4.3 Lancretγ = τ ± k. Then a Legendre curve on S 3 satisfies 4.4 ks = + γ + γ γ γ γ γ + γ + + γ γ 3, τs =. Now, recall that E 3 is a Killing vector field; then our notion of a slant curve belongs to the class of general helices introduced in [], page 55. For the case of S 3 the Lancret theorem from [], page 56, yields the same Sasakian Lancret invariant τ ± /k For m > and l = we have M m = S 4m R If m < and l = then we have Mm 3 = H 4m R where H k is the hyperbolic plane of constant Gaussian curvature k <. For both the Cases 4.4 and 4.5 a slant curve has ks = γ γ γ γ + m sin γ γ γ γ 4.5, 4.6 F sin θ F τs = γ γ γ γ F sin θ + mγ γ γ γ F cos θ, which means that the Legendre curves in these ambient manifolds have vanishing torsion If m > and l we get SU\{ }. The Legendre curves of this manifold have a constant non-null torsion. The general properties of the curves in SU are studied in [] If m < and l we have SL, R. In the following, returning to the general case we use along γ the cylindrical coordinates xs = rs cos βs, ys = rs sin βs, zs = zs and then 3. becomes 4.7 r + r β = [ + mr ] sin θ, 3

14 which yields 4.8 β = [ + mr ] sin θ r r. Now, we make the choice of a positive real constant r = r and then 4.9 βs = ± + mr sin θ r s. We get the final expression of the curve for the positive sign in 4.9: 4. γ r = r cos This curve is a helix with 4. + mr sin θ r s, r sin + mr sin θ k = + mr sin θ l sin θ cos θ, r τ = l + + mr sin θ l cos θ cos θ. For Examples 4.4 and 4.5 this yields helices with r r s, 4. k = + mr sin θ r, τ = + mr sin θ cos θ r. cos θ + lr s. We can use the relation 4. in order to obtain slant curves with a prescribed curvature for the case 4.5. Proposition 4.. Let c > be given and m = m <. Then on H 4 m R the curve 4.7 with 4.3 r = c + 4 m sin 4 θ c m sin θ is a helix θ-slant curve with k = c and τ = c cos θ/sin θ. P r o o f. The curve is γ r = cs cs r cos, r sin, s cos θ sin θ sin θ and a simple computation yields the conclusion. Acknowledgement. We are very gratefully to Professor Aurel Bejancu from Kuweit University for several improvements in a preliminary version. 4

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16 [] Z. Olszak: Normal almost contact metric manifolds of dimension three. Ann. Pol. Math , 4 5. [3] Y.-L. Ou, Z.-P. Wang: Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries. J. Geom. Phys. 6, [4] P. Piu, M. M. Profir: On the three-dimensional homogeneous SO-isotropic Riemannian manifolds. An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 57, [5] J. Van Der Veken: Higher order parallel surfaces in Bianchi-Cartan-Vranceanu spaces. Result. Math. 5 8, [6] J. We lyczko: On Legrende curves in 3-dimensional normal almost contact metric manifolds. Soochow J. Math. 33 7, [7] J. We lyczko: On Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Result. Math. 54 9, Authors addresses: C o n s t a n t i n C ă l i n, Technical University Gh. Asachi, Iaşi, Romania, cnstc@yahoo.com; M i r c e a C r a s m a r e a n u, University Al. I. Cuza, Iaşi, Romania, mcrasm@uaic.ro. 6