GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE. 1. Introduction

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1 ACTA MATHEMATICA VIETNAMICA 205 Volume 29, Number 2, 2004, pp GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE HANDAN BALGETIR AND MAHMUT ERGÜT Abstract. In this paper, we define (r+1)-dimensional generalized null scrolls in the n-dimensional Lorentzian space R n 1 and examine their geometric invariants and characteristic properties. 1. Introduction Ruled surfaces have an important role in Differential Geometry. The (r + 1)- dimensional generalized ruled surfaces in the n-dimensional Euclidean space E n are studied by Juza [7], Frank and Giering [4] and Thas [10]. Some properties of 2-dimensional ruled surfaces are given by Thas [11]. In recent years, the semiruled surfaces and their curvatures have been studied in the semi-euclidean space (see [3]). However, these work constructed the generalized ruled surfaces as bases on spacelike curves or timelike curves in the semi-euclidean space. Null curves have many properties very different from spacelike or timelike curves and they are very interesting and important in Differential Geometry (see [2]). Graves [5] first introduced the notion of B-scroll as bases on a null curve and a null line in the 3-dimensional Lorentzian space E1 3. Eν n+1 In this paper, we introduce the notion of (r + 1)-dimensional generalized null scrolls in the n-dimensional Lorentzian space and study their characteristic properties. To do this, we use a general Frenet equations of null curves and pseudoorthonormal basis. We also obtaine curvatures of generalized null scrolls in the n-dimensional Lorentzian space R n 1. Let M be an m-dimensional Lorentzian submanifold of R1 n. Let be a Levi- Civita connection of R1 n and a Levi-Civita connection of M. If X, Y χ(m) and h is the second fundamental form of M, then we have the Gauss equation (1.1) (1.2) X Y = X Y + h(x, Y ). Let ξ be a unit normal vector field on M. Then the Weingarten equation is X ξ = A ξ (X) + Xξ, Received January 27, Mathematics Subject Classification. 53B30. Key words and phrases. Generalized null scroll, pseudo-orthogonal trajectory, pseudoorthonormal frame.

2 206 HANDAN BALGETIR AND MAHMUT ERGÜT where A ξ determines at each point a self-adjoint linear map on T p (M) and is a metric connection on normal bundle of M. In this paper, A ξ will be used for the linear map and the corresponding matrix of the linear map. From the equations (1.1) and (1.2), we have (1.3) and (1.4) Also by the equations (1.3) and (1.4), (1.5) X Y, ξ = h(x, Y ), ξ X Y, ξ = A ξ (X), Y. h(x, Y ), ξ = A ξ (X), Y. Let {ξ 1, ξ 2,..., ξ n m } be an orthonormal basis of χ (M). Then there exist smooth functions h j (X, Y ), j = 1,..., n m, such that (1.6) h(x, Y ) = n m h j (X, Y )ξ j and furthermore we may define the mean curvature vector field H by (1.7) H = n m tracea ξj m ξ j. If H(p) = 0 for each p M, then M is said to be minimal [9]. Let ξ be a unit normal vector, then the Lipschitz-Killing curvature in the direction ξ at the point p M is defined by (1.8) The Gauss curvature is defined by (1.9) G(p, ξ) = det A ξ (p). G(p) = n m G(p, ξ j ) and if G(p) = 0 for all p M, we say that M is developable. In particular, if the Lipschitz-Killing curvature is zero for each point and each normal direction, then M is developable [11]. Following [6], we define M(A) for any matrix A = [a ij ] by M(A) = i,j (a ij ) 2. Let {ξ 1, ξ 2,..., ξ n m } be an orthonormal basis of χ (M). normal curvature K N of M is defined by (1.10) K N = n m i, M(A ξi A ξj A ξj A ξi ). Then the scalar

3 GENERALIZED NULL SCROLLS 207 Let M be a Lorentzian manifold. For every X, Y, Z, W χ(m), the 4 th. order covariant tensor field (1.11) R(X, Y, Z, W ) = R(Z, W )Y, X is called the Riemannian-Christoffel curvature tensor field and its value at a point p M is called the Riemannian-Christoffel curvature of M at p. The Riemann curvature at p M is denoted by (1.12) K(P ) = R(X, Y )Y, X p. From the equations (1.1) and (1.11) we get (see [9]) (1.13) R(X, Y )Y, X = h(x, X), h(y, Y ) h(x, Y ), h(x, Y ). Let M be an m-dimensional Lorentzian manifold and P a null plane of T p (M). Then as a real number, K U (P ) defined by K U (P ) R(W, U)U, W (1.14) p =, p M W, W p is called the null sectional curvature of P with respect to U, where W is an arbitrary non-null vector in P and U is a null vector of T p (M) [2]. Let M be an n-dimensional Lorentzian manifold and R the Riemann curvature tensor. The tensor field Ric defined by n (1.15) Ric(X, Y ) = ε i R(e i, X)Y, e i is called the Ricci curvature tensor field, where {e 1,..., e n } is a system of orthonormal basis of T p (M) and the value of Ric(X, Y ) at p M is called the Ricci curvature. Let M be an n-dimensional Lorentzian manifold and {e 1,..., e n } an orthonormal basis of T p (M) at p M. The scalar curvature of M is defined by (1.16) or (1.17) r = r = n n ε i Ric(e i, e i ) n ε i ε j R(e j, e i )e i, e j, where ε i = e i, e i so that ε i = { 1, if e i timelike, 1, if e i spacelike (see [1]).

4 208 HANDAN BALGETIR AND MAHMUT ERGÜT A basis {X, Y, Z 1,..., Z n 2 } of an n-dimensional Lorentzian space R1 n a pseudo-orthonormal basis if the following conditions are fullfilled: is called (1.18) X, X = Y, Y = 0, X, Y = 1, X, Z i = Y, Z i = 0 for 1 i n 2, Z i, Z j = δ ij, for 1 i n 2 (see [8]). 2. Null curves In this section, we recall the notion of null curve in the Lorentzian manifold [2]. Let (M,, ) be a real (n+2)-dimensional Lorentzian manifold and α a smooth null curve in M locally given by α i = α i (t), t I R, i {1,..., n + 2} for a coordinate neighbourhood U on α. Then the tangent vector field on U satisfies the condition dα ( dα 1 dt = dt,..., dαn+2 ) dt dα dt, dα = 0. dt We denote by T α the tangent bundle of α and T α is defined as follows T α = p α T p α ; T p α = {v p T p (M); v p, ξ p = 0}, where ξ p is null vector tangent at any p α. Clearly, T α is a vector bundle over α of rank (n + 1). Since ξ p is null, it follows that the tangent bundle T α is a vector subbundle of T α, of rank 1. Suppose S(T α ) is the complementary vector subbundle to T α in T α, i.e., T α = T α S(T α ), where means the orthogonal direct sum. It follows that S(T α ) is a nondegenerate n-dimensional vector subbundle of T M. We call S(T α ) a screen vector bundle of α. We have (2.1) T M α = S(T α ) S(T α ), where S(T α ) is a 2-dimensional complementary orthogonal vector subbundle to S(T α ) in T M α [2].

5 GENERALIZED NULL SCROLLS 209 Theorem 2.1. [2]. Let α be a null curve of a semi-riemannian manifold (M,, ) and S(T α ) a screen vector bundle of α. Then there exists a unique vector bundle E over α of rank 1 such that on each coordinate neighbourhood U α there is a unique section N Γ(E U ) satisfying dα (2.2) dt, N = 1, N, N = N, X = 0, for every X Γ(S(T α ) U ). Now, suppose α is a null curve of an (n + 2)-dimensional Lorentzian manifold (M,, ). Denote by the Levi-Civita connection on M and dα X. Then the dt following equations can be obtained: (2.3) X X = λx + k 1 Z 1, X Y = λy k 2 Z 1 k 3 Z 2, X Z 1 = k 2 X + k 1 Y + k 4 Z 2 + k 5 Z 3, X Z 2 = k 3 X k 4 Z 1 + k 6 Z 3 + k 7 Z 4, X Z n 1 = k 2n 3 Z n 3 k 2n 2 Z n 2 + k 2n Z n, X Z n = k 2n 1 Z n 2 k 2n Z n 1 provided n 5, where λ and {k 1,..., k 2n } are smooth functions on U and {Z 1,..., Z n } is a certain orthonormal basis of Γ(S(T α ) U ). We call F = {X, Y, Z 1,..., Z n } a Frenet frame. It is also called a pseudo-orthonormal frame since the equations (1.18) holds on M along α with respect to the screen vector bundle S(T α ). The functions {k 1,..., k 2n } are called curvature functions of α with respect to F and the equations (2.3) are called the Frenet equations with respect to F [2]. 3. Generalized null scrolls in R n 1 Let α : I R R1 n be a smooth null curve in the n-dimensional Lorentzian space R1 n and {X, Y, Z 1,..., Z n 2 } be a pseudo-orthonormal frame along the null curve α. Let {Y (t), Z 1 (t),..., Z r 1 (t)} be a null basis defined at each point α(t) of the null curve α. This system spans a subspace of the tangent space T α(t) (R1 n) at α(t) R1 n. This space is denoted by W r(t). It is a r-dimensional subspace of the form W r (t) = Sp{Y (t), Z 1 (t),..., Z r 1 (t)} R n 1. W r (t) will be called a degenerate subspace and the following equalities are satisfied: Y (t), Y (t) = 0, Z i (t), Z j (t) = δ ij, Y (t), Z i (t) = 0

6 210 HANDAN BALGETIR AND MAHMUT ERGÜT for 1 i, j r 1. Definition 3.1. Let α be a null curve in the n-dimensional Lorentzian space R1 n. While the r-dimensional degenerate subspace W r (t) moves along a null curve α in R1 n, it forms an (r +1)-dimensional surface. This is called the (r +1)-dimensional generalized null scroll in the n-dimensional Lorentzian space R1 n and denoted by M. Then M can be expressed by the parametric equation (3.1) Ψ : I X R r R n 1, r 1 (t, u) Ψ(t, u) = α(t) + u 0 Y (t) + u i Z i (t) where u = (u 0, u 1,..., u r 1 ). Note that ( rank(ψ t, Ψ u0,..., Ψ ur 1 ) = rank α (t) + u 0 Y (t) r 1 ) + u i Z i(t), Y (t), Z 1 (t),..., Z r 1 (t) = r + 1. It is easy to check that M is a Lorentzian submanifold. Definition 3.2. W r (t) is called the generating space (or generating degenerate space) at the point α(t) of the (r + 1)-dimensional generalized null scroll M and the null curve α is called the base curve of M. Definition 3.3. Let M be an (r + 1)-dimensional generalized null scroll in R n 1. The subspace R(t) = Sp{Y (t), Z 1 (t),..., Z r 1 (t), Y (t), Z 1(t),..., Z r 1(t)} is said to be the asymptotic bundle of the generalized null scroll M. Definition 3.4. Let M be an (r + 1)-dimensional generalized null scroll in R n 1. If there exists a timelike or spacelike curve such that it meets perpendicularly to each one of the generating spaces, then this curve is called an orthogonal trajectory of the generalized null scroll M. Definition 3.5. Let M be an (r + 1)-dimensional generalized null scroll in R n 1 and W r (t) be generating space of M. If there exists a null curve β on M such that for each t I, (3.2) β (t), Y (t) = 1, β (t), Z i (t) = 0, 1 i r 1, then the null curve β is called a pseudo-orthogonal trajectory of the generalized null scroll M. Thus we have following theorem:

7 GENERALIZED NULL SCROLLS 211 Theorem 3.1. Let M be an (r + 1)-dimensional generalized null scroll in R1 n and W r (t) be generating space of M and α(t) be base curve of M. Then there exists a pseudo-orthogonal trajectory if and only if r 1 (3.3) u i Z i, Y = 0 and u i = µ u 0 (t)dt + C, where µ = Y, Z i, i = 1,..., r 1; u 0, u 1...., u r 1 R. Proof. It can be easily derived from the equation (3.2). 4. On the curvatures of generalized null scroll Let M be an (r +1)-dimensional generalized null scroll and α(t), t I the base curve of M. Let {Y (t), Z 1 (t),..., Z r 1 (t)} be a null basis of the generating space W r (t). Let us choose the base curve α to be a pseudo-orthogonal trajectory of the generating spaces W r (t). Then M is given by (4.1) r 1 Ψ(t, u 0, u 1,..., u r 1 ) = α(t) + u 0 Y (t) + u i Z i (t), ( ) where (u 0, u 1,..., u r 1 ) R. Let us choose X = Ψ such that {X, Y, Z 1,..., t Z r 1 } is a pseudo-orthonormal basis of the space of vector fields χ(m). By (4.1) and (1.1), we have (4.2) (4.3) Zi Z j = 0, Y Z i = 0, Zi Y = 0, Y Y = 0, h(z i, Z j ) = 0, h(z i, Y ) = 0, 1 i, j r 1. This means that the generating space W r (t) is totally geodesic. Also, we get (4.4) (4.5) Zi X = h(z i, X), 1 i r 1 Y X = h(y, X). Theorem 4.1. Let M be an (r + 1)-dimensional generalized null scroll in R n 1. Consider the pseudo-orthonormal basis {X, Y, Z 1,..., Z r 1 } in a neighbourhood of a point p of M. Then the null sectional curvature K X (P ) in the two-dimensional direction null plane P of M spanned by the vectors X p and (Z i ) p is given by (4.6) K X (P ) = Zi X, Zi X p, 1 i r 1. Proof. From the equation (1.14), we can write K X (P ) = R(Z i, X)X, Z i Z i, Z i Since {X, Y, Z 1,..., Z r 1 } is a pseudo-orthonormal basis, we have (4.7) K X (P ) = R(Z i, X)X, Z i. Also, using the equations (1.13), (4.3) and (4.4) for p M, we get (4.8) K X (P ) = K(Z i, X) = Zi X, Zi X p, 1 i r 1,

8 212 HANDAN BALGETIR AND MAHMUT ERGÜT and this completes the proof. Theorem 4.2. Let M be an (r+1)-dimensional generalized null scroll in R n 1 and {X, Y, Z 1,..., Z r 1 } is a pseudo-orthonormal basis in a neighbourhood of a point p of M. Then the sectional curvature in the two-dimensional direction Lorentzian plane σ of M spanned by {X p, Y p } p M, given by (4.9) K(σ) = K(X, Y ) = Y X, Y X p. Proof. It can be similarly proved as Theorem 4.1. Corollary 4.1. An (r + 1)-dimensional generalized null scroll M in R1 n is developable if and only if each sectional curvatures of M is identically zero, i.e., K(X, Z i ) K(X, Y ) 0, 1 i r 1. Proof. It is obvious from the equations (1.12), (1.13) and (4.3). Suppose that the vector field system {ξ 1, ξ 2,..., ξ } is an orthonormal basis of T p M at p M. Then {X, Y, Z 1,..., Z r 1, ξ 1, ξ 2,..., ξ } is a pseudo-orthonormal basis of T p (R1 n ) at p M. Thus, the equations of the derivative can be written as follows: r 1 X ξ j = a j 00 X + bj 00 Y + c j 0t Z t + e j 0q ξ q, (4.10) t=1 t=1 q=1 r 1 Y ξ j = a j 10 X + bj 10 Y + c j 1t Z t + e j 1q ξ q, t=1 q=1 r 1 Zi ξ j = a j 1i X + bj 1i Y + d j it Z t + f j iq ξ q, where 1 i r 1, 1 j n r 1. If we consider the Weingarten equation (1.2) and the equations (4.10), then the matrix A ξj that corresponds to the linear mapping is a j 00 b j 00 c j 01 c j 02 c j 0(r 1) 0 b j b j A ξj = (4.11) 0 b j b j 1(r 1) and this means that deta ξj = 0 if r > 1, from which we obtain following corollary. Corollary 4.2. If r > 1, then the Lipschitz-Killing curvature of the (r + 1)- dimensional generalized null scroll M is zero at each point in each normal direction. q=1

9 GENERALIZED NULL SCROLLS 213 Theorem 4.3. Let M be an (r + 1)-dimensional generalized null scroll in R n 1 and {X, Y, Z 1,..., Z r 1 } a pseudo-orthonormal basis of χ(m). Then the Ricci curvature of M is in the direction of the vector fields Y and Z j, 1 i r 1, satisfies, respectively, (4.12) (4.13) Ric(Y, Y ) = (b j 10 )2, Ric(Z i, Z i ) = (b j 1i )2, where b j 10 and bj 1i are the components of the matrix A ξ j. Proof. Using the equations (1.3), (1.4), (1.13) and (1.15), we get the equations (4.12) and (4.13). Corollary 4.3. The Ricci curvature of the (r + 1)-dimensional generalized null scroll M in R1 n in the direction of the vector field X is given by (4.14) r 1 Ric(X, X) = Ric(Z i, Z i ) + Ric(Y, Y ). Theorem 4.4. Let M be an (r + 1)-dimensional generalized null scroll in R n 1 and {X, Y, Z 1,..., Z r 1 } a pseudo-orthonormal basis of χ(m). Then the scalar curvature of M is equal to twice Ricci curvature in the direction of the vector field X. Proof. By the equation (1.16) the scalar curvature of M can be expressed by Using Corollary 4.3 we obtain (4.15) This completes the proof. r 1 r = Ric(X, X) + Ric(Y, Y ) + Ric(Z i, Z i ). r = 2Ric(X, X). By (4.12), (4.13), (4.14), (4.15), we obtain the following corollary. Corollary 4.4. The scalar curvature of M is given by r 1 (4.16) r = (b j 1i )2 + (b j 10 )2 Theorem 4.5. Let M be an (r + 1)-dimensional generalized null scroll in R1 n and {X, Y, Z 1,..., Z r 1 } be a pseudo-orthonormal basis of χ(m) and X be the tangent vector of the base curve of M. Then the mean curvature of M is (4.17) H = 2 h(x, Y ). r + 1.

10 214 HANDAN BALGETIR AND MAHMUT ERGÜT Proof. By the equations (4.4), (4.5), (4.10) and (4.11) we see that (4.18) (4.19) Therefore, we can write (4.20) h(x, Y ) = b j 10 ξ j, trace(a ξj ) = 2b j 10. h(x, Y ) = 1 2 Thus, from the equation (1.7) we obtain (trace A ξj )ξ j. H = 2 h(x, Y ). r + 1 Hence we get following corollary. Corollary 4.5. (r + 1)-dimensional generalized null scroll M is minimal if and only if h(x, Y ) = 0. Theorem 4.6. Let {Y, Z 1,..., Z r 1 } be a null basis for the generating space of an (r+1)-dimensional generalized null scroll M and X be a tangent vector field to the base curve. Suppose that the pseudo-orthogonal trajectory of the generating space is the base curve of M. Then the minimal generalized null scroll M is totally geodesic if and only if X is an asymptotic vector field and the conjugate to each vector field Z i, 1 i r 1. Proof. Let {X, Y, Z 1,..., Z r 1 } be a pseudo-orthonormal basis of χ(m) and for each U, V χ(m) we can write and from now (4.21) r 1 U = a 0 X + b 0 Y + b i Z i, r 1 V = a 1 X + c 0 Y + c j Z j h(x, Y ) = a 0 a 1 h(x, X) + (a 0 c 0 + b 0 a 1 )h(x, Y ) + b 0 c 0 h(y, Y ) r 1 r 1 + (a 0 c i + a 1 b i )h(x, Z i ) + (b 0 c i + c 0 b i )h(y, Z i ) + r 1 i, b i c i h(z i, Z j ).

11 GENERALIZED NULL SCROLLS 215 If M is totally geodesic, then h is identically zero [2]. So we have h(x, X) = 0, h(x, Z i ) = 0, 1 i r 1. This means that X is an asymptotic vector field and conjugate to each vector field Z i [2]. If h(x, X) = 0 and h(x, Z i ) = 0, 1 i r 1, then from the equation (4.21) we find h(x, Y ) = 0 and this completes the proof. From (4.18), (4.19) and (4.20), we obtain (4.22) (r + 1) 2 H 2 = 4 (b j 10 )2. Theorem 4.7. The scalar normal curvature of an (r +1)-dimensional null scroll M is given by (4.23) K N = (r + 1) i, + 2 i, H 2 r 1 r 1 (c i 0t) 2 (r (r + 1)2 H 2)( (r + 1) 2 H 2) 4 t=1 ( (c i 0t b j 1t )2 c i 0tb j 1t cj 0t bi 1t c j 0t bi 10c i 0tb j 10 bi 1tb j 10 bj 1t bi 10) t=1 r 1 t k=1 (c i 0tb j 1t ci 0k bj 1k + ci 0tb j 1t cj 0k bi 1k ), where H and r are the mean curvature vector field and scalar curvature of M, respectively and c j 0t, bj 10, bj 1t, 1 t r 1, 1 j n r 1 are the elements of the matrix A ξj. Proof. It can be easily proved from the equation (1.10) and by the some calculations. Corollary 4.6. The scalar normal curvature of a minimal (r + 1)-dimensional generalized null scroll is given by (4.24) K N = 2 { r 1 i, + i, [(c i 0tb j 1t )2 c i 0tb j 1t cj 0t bi 1t c j 0t bi 10c i 0tb j 10 bi 1tb j 10 bj 1t bi 10] t=1 r 1 t k=1 (c i 0tb j 1t ci 0k bj 1k + ci 0tb j 1t cj 0k bi 1k ) } Proof. If M is minimal, H = 0 and so the corollary is clear. Corollary 4.7. If the generalized null scroll M is minimal and totally geodesic, then the scalar normal curvature of M is identically zero.

12 216 HANDAN BALGETIR AND MAHMUT ERGÜT Proof. Since M is minimal and totally geodesic, A ξj is the zero map for each j = 1,..., n r 1. So the scalar normal curvature of M is identically zero. This completes the proof. References [1] M. Bektas, On the curvatures of (r + 1)-dimensional generalized time-like ruled surface, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 19 (2003), [2] K. L. Duggal, and A. Bejuancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer, Dordrecht, [3] C. Ekici, and A. Görgülü, On the curvatures of (k + 1)-dimensional semi-ruled surfaces in E n υ, Math. Comp. Appl. 5 (2000), no. 3, [4] H. Frank, and O. Giering, Verallgemeinerte Regelflachen, Mathematische Zeitschrift 150 (1976), [5] L. K. Graves, Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), [6] C. S. Houh, Surfaces with maximal Lipschitz-Killing curvature in the direction of mean curvature vector, Proc. Amer. Math. Soc. 35 (1972), [7] M. Juza, Ligne de striction sur une generalisation a plusierurs dimensions d une surface regleé, Czechosl. Math. J. 12 (1962), [8] M. A. Magid, Lorentzian isoparametric hypersurfaces, Pacific J. Math. 118 (1985), no. 1, [9] B. O Neill, Semi-Riemannian Geometry, Academic Press, New York, [10] C. Thas, Minimal monosystems, Yokohama Math. J. 26 (1978), no. 2, [11] C. Thas, Properties of ruled surfaces in the Euclidean space E n, Bull. Inst. Math. Acad. Sinica 6 (1978), Department of Mathematics Firat University Elazığ/TÜRKİYE address: hbalgetir@firat.edu.tr