is constant [3]. In a recent work, T. IKAWA proved the following theorem for helices on a Lorentzian submanifold [1].

Transcription

1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.2. ON GENERAL HELICES AND SUBMANIFOLDS OF AN INDEFINITE RIEMANNIAN MANIFOLD BY N. EKMEKCI Introduction. A regular curve in an indefinite Riemannian manifold whose third curvature is zero and the first, two and k 2 are constant is called a helix. It is called general helix if the latter condition is replaced with a weaker one, namely that k 2 is constant [3]. In a recent work, T. IKAWA proved the following theorem for helices on a Lorentzian submanifold []. Theorem. Let M (dim M 3) be a connected Lorentzian submanifold of an indefinite Riemannian manifold M i. If for some, k 2 > 0, every timelike helix with curvatures and k 2 in M is a timelike helix in M i, then M is a totally geodesic submanifold in M i. In this paper, we generalize this theorem to the case of a general helix.. Preliminaries. Let M γ be an n-dimensional indefinite Riemannian manifold of index γ (0 γ n) isometrically immersed into an m- dimensional indefinite Riemannian manifold M i of index i. Then M γ is called an indefinite Riemannian submanifold of M i. Especially if γ=, then M is called a Lorentzian submanifold of M i. We denote the metrics of M γ and M i by the symbol, and the covariant differentiation of M γ (resp. M i ) by D (resp. D). Then we have the Gauss formula () D X Y = D X Y + B(X, Y )

2 264 N. EKMEKCI 2 where X and Y are tangent vector fields of M γ and B is the second fundamental form of M γ. Weingarten s formula is (2) D X ξ = A ξ X + D Xξ where X (resp. ξ) is a tangent (resp. normal) vector fields of M γ. D is the covariant differentiation with respect to the induced connection in the normal bundle N(M γ ) and A ξ is the shape operator of M γ. We have the following relation (3) A ξ X, Y = B(X, Y ), ξ. For the second fundamental form and the shape operator, we define their covariant derivatives by (4) DB(X, Y, Z) = D Z B(X, Y ) B(D Z X, Y ) B(X, D Z Y ) (5) D 2 B(X, Y, Z, W ) = D W ( DB(X, Y, Z)) DB(D W X, Y, Z) DB(X, D W Y, Z) DB(X, Y, D W Z) (6) ( D Y A) ξ X = D Y (A ξ X) A D Y ξ X A ξ D Y X where X, Y, Z, W are tangent vector fields of M γ and ξ is a normal vector field of M γ. The mean curvature vector field H of M γ is defined by (7) H = n n e i, e i B(e i, e i ) i= where e, e 2,..., e n } is an orthonormal frame of M γ. H is said to be parallel when D H = 0 holds. If the second fundamental form B satisfies B(X, Y ) = X, Y H for all vector field X, Y of M γ, then M γ is called a totally umbilical submanifold. If the second fundamental form vanishes identically on M γ, then M γ is said to be totally geodesic.

3 3 ON GENERAL HELICES AND SUBMANIFOLDS 265 Since the second fundamental form B is a bilinear symmetric function on T p (M γ ), we have the following Lemmas []. Lemma.. For any point p of M, we assume that B satisfies B(t, s) = 0, where t T p (M ) is a unit timelike vector and s T p (M ) is a unit spacelike vector such that t, s = 0. Then M is a totally umbilical submanifold. Lemma.2. Let H be the mean curvature vector field of a Lorentzian submanifold M. For any point p of M, we assume that H satisfies D s H = 0, for any spacelike vector s T p (M ). Then H is parallel. Lemma.3. [2] For a totally umbilical submanifold M γ, the following conditions are equivalent: () D H = 0, (2) DB(X, X, X) = 0 for any X T p (M γ ). 2. Helices on an Indefinite Riemannian Manifold. Let α(t) be a regular curve on an indefinite Riemannian manifold M γ. We denote the tangent vector field α (t) = X. When X, X =, α is called a unit speed curve. In this paper, a unit speed curve α in M γ will be called a general helix if the third curvature of it is zero and the ratio of the first two constant. Let α(t) be a unit speed timelike curve in M γ. If the principal vector field Y and the binormal vector field Z are spacelike, then we have the following Frenet formulas along α(t) : (8) D X X = (t)y D X Y = (t)x + Z D X Z = Y where D denotes the covariant differentiation in M γ. Theorem 2.. A unit speed timelike curve α satisfying (8) on M γ is a general helix if and only if (9) D 3 XX 3k D 2 XX (k ) 3(k ) 2 + k 2 k2 2 D X X = 0 k 2 k 2 is

4 266 N. EKMEKCI 4 Proof. Suppose that α(t) is a general helix. Then (9) is clear from (8). Conversely let us assume that (9) holds. We show that the curve α(t) is a general helix. D X X X. Let be Y of length such that D X X = (t)y. If (t) = 0, α is a geodesic. If not, let D X Y = (t)x + Z, with Z of length (Y, Z are space like).if = 0, α is a circle. If not, let us compute D X Z. We can write Z = (t) D XY (t) X and so, [ D X Z = (D 3 (t) XX (t) 3k (t) D2 XX ) ] (k (t) (t) 3(k (t)) 2 k 2(t) + k(t) 2 k2(t) 2 D X X ) [((k (t)) (t) 2k DXX 2 (t) ( ((t)) 2 = k 2 (t) and so k 2 (t) (t) (t) ) ] ( (t)k 2(t) k (t) D X X (t) ( (t) + k2(t) 2 k ( ) k (t) D 2 XX D X Z = Y Thus α(t) is a general helix. (t) + k k 3(t) ) X = ( ) k (t) D X X ( ) k (t) = 0 (t) = constant Another expression of equation (9) is shown below, ) ( ) k (t) X (0) D X D X D X X KD X X = 3k D X Y K = k + k 2 k 2 2

5 5 ON GENERAL HELICES AND SUBMANIFOLDS 267 Theorem 2.2. Let M (dim M 3) be a connected Lorentzian submanifold of an indefinite Riemannian manifold M i. If every timelike general helix with curvatures and k 2 in M is a timelike general helix in M i, then M is a totally geodesic submanifold in M i. Proof. For any point p of M, let x, y and z be three orthonormal vectors in T p (M ) such that x is timelike, y and z are spacelike, respectively. Let α(t) be a timelike general helix satisfying (8) on M such that α(0) = p, α (t) = X, X(0) = x, Y (0) = y, Z(0) = z (D X X)(p) = y, (D X Y )(p) = x + k 2 z, (D X Z)(p) = k 2 y where X, Y and Z is the tangent, the principal, the binormal vector field of α respectively. From Theorem 2., X satisfies (0). D X D X D X X KD X X = 3k D X Y, K = k + k 2 k 2 2 Since α(t) is a general helix on M i, we have () D X D X D X X K D X X = 3k D X Y where K = k k2. Substituting (), (2) into this equation, we obtain for the normal part of M ( ) 2 } k 3 K B(X, X) 9 B(X, Y ) 3 DB(X, X, X)+ (2) (3) +4B(X, D X D X X) + 5 DB(X, D X X, X) + DB(X, X, D X X)+ + D 2 B(X, X, X, X) + 3B(D X X, D X X) B(X, A B(X,X) X) = 0 and for the tangent part of M ( ) 2 ( ) } k K K + 3 k 2 3 D X X + 3 k 3 k } D X D X X+ +3 k A B(X,X) X D X A B(X,X) X 2A DB(X,X,X) X 5A B(X,DX X)X A B(X,X) D X X = 0.

6 268 N. EKMEKCI 6 By (2) the following equality holds good. 3 ( k ) 2 K k 2 } B(X, X) 9 } + k B(X, Y ) (4) 2k 2 B(Y, Y ) k 2 B(X, Z) 3 k DB(X, X, X)+ +5D XB(X, Y ) + DB(X, X, D X X)+ + D 2 B(X, X, X, X) B(X, A B(X,X) X) = 0. Replacing Z by Z in (4) we have B(X, Z) = 0, where X and Z are orthonormal vectors of T p (M ). We see that M is totally umbilical by virtue of Lemma. (5) DB(X, X, D X X) = DB(X, X, k, Y ) = = D Y B(X, X) 2B(D Y X, X) M is totally umbilical (6) B(D Y X, X) =< D Y X, X > H = 0 (7) D Y B(X, X) =< X, X > D Y H (8) DB(X, X, k Y ) = < X, X > D Y H (9) 3 ( k ) 2 K k 2 } < X, X > H 2k 2 < Y, Y > 3 k DB(X, X, X) + k < X, X > D Y H+ + D 2 B(X, X, X, X) B(X, A B(X,X) X) = 0 Replacing Y by Y in (9) and using the fact that M is totally umbilical we obtain D Y H = 0.

7 7 ON GENERAL HELICES AND SUBMANIFOLDS 269 Hence from Lemma.2 we see that the mean curvature vector field is parallel. From Lemma.3 DB(X, X, X) = 0 and D 2 B(X, X, X, X) = 0 for a timelike vector X, which imply that (2) is reduced to (20) K 3 ( k ) 2 k 2 H, H } H = 0. (2) On the other hand, the inner product of (3) with Y gives ( ) 2 ( ) } k K K + 3 k 2 3 < D X X, Y > k 3 k } < D X D X X, Y > +3 < A B(X,X) X, Y > < D X A B(X,X) X, Y > < A B(X,X) D X X, Y >= 0 Since M is totally umbilical with parallel mean curvature vector, this equation is reduced to ( ) 2 k (22) K K = H, H. Combining equation (22) with equation (20), we have (23) K k 2 3 k } H = 0. Hence, H = 0. This means that M is a totally geodesic submanifold of M i. Acknowledgement. The author would like to express his hearty thanks to Professor M. Anastasiei for his various advices. REFERENCES. IKAWA, T. On Curves and Submanifolds in an Indefinite Riemannian Manifold, Tsukaba J. Math. 9 (985),

8 270 N. EKMEKCI 8 2. ABE N., NAKANISHI, Y. and YAMAGUCHI S. Circles and Spheres in Pseudo- Riemannian Geometry, Aequationes Mathematicae, 39 (990), MILLMAN R.S. and PARKER, G.D. Elements of Differential Geometry, Prentice Hall, Englewood Cliffs, New Jersey IKAWA, T. On Some Curves in Riemannian Geometry, Soochow J. Math. 7 (980) O NEIL, B. Semi Riemannian Geometry, Academic Press, New York, HICKS N. Submanifolds of Semi Riemannian Manifolds, Rend. Circ. Mat Polermo, 2 (963), NOMIZU, K. and YANO, K. On Circles and Spheres in Riemannian Geometry, Math. Ann, 20 (974), Received:.VI.999 Ankara University Faculty of Sciences Department of Mathematics 0600, Tandogan, Ankara TURKEY ekmekci@science.ankara.edu.tr