Curves of Restricted Type in Euclidean Spaces

Submanifolds of restricted type were introduced in [7]. In the present study we consider restricted type of curves in E. We give some special examples. We also show that spherical curve in S(r) ⊂ E is of restricted type if and only if either f(s) is constant or a linear function of s of the form f(s) = ±s+ b and every closed W − curve of rank k and of length 2πr in E is of restricted type.


Introduction
Let M n be an n−dimensional submanifold of a Euclidean space E m . Let x,H and ∆ respectively be the position vector field, the mean curvature vector field and the Laplace operator of the induced metric on M n . Then, as is well known (see e.g. [2]) (1) ∆x = −nH, which shows, in particular, that M n is a minimal submanifold in E m if and only if its coordinate functions are harmonic (i.e. they are eigenfunctions of ∆ with eigenvalue 0). As a generalization of T. Takahashi's condition and following an idea of O. Garay [13], some of the authors together with J. Pas [10] initiated the study of submanifolds M n in E m such that (2) ∆x = Ax + B for some fixed vector B ∈ E m and a given matrix A ∈ R m×m . This study was continued by the first author together with M . Petrovic [5] and independently by T. Hasanis and T. Vlachos [14]. During the study of submanifolds of R m satisfying (2), it was observed that all these matrices A p are equal for all p ∈ M , or equivalently there exists a fixed matrix A ∈ E m×m (determining, of course, a linear endomorphism of E m ) such that for all p ∈ M and for all X ∈ T p M , A H X = (AX) T .
As the relation (3) expresses a strong relationship between differential geometry and linear algebra, we do think it would be worthwhile to study submanifolds satisfying this condition; such submanifolds are said to be of restricted type.
Submanifolds of restricted type were introduced in [7] by the author B.Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken. The class of submanifolds of restricted type is large which includes 1−type submanifolds, pseudoumbilical submanifolds with constant mean curvature, submanifolds satisfying either Gray's condition or Dillen Pas Verstraelen's condition, all k−type curves lying fully in E 2k , all null k−type curves lying fully in E 2k−1 , the products of submanifolds of restricted type, the diagonal immersions of restricted type submanifolds and equivariant isometric immersions of compact homogeneous spaces. In [7], it is shown that a hypersurface of restricted type is either minimal, or a part of the product of a sphere and a linear subspace, or a cylinder on a plane curve of restricted type, and all planar curves of restricted type are classified.

Basic Concepts
In the present section we recall definitions and results of [1]. Let x : M → E m be an immersion from an n−dimensional connected Riemannian manifold M into an m−dimensional Euclidean space E m . We denote by g the metric tensor of E m as well as the induced metric on M . Let∇ be the Levi-Civita connection of E m and ∇ the induced connection on M . Then the Gaussian and Weingarten formulas are given, respectively, by where X, Y are vector fields tangent to M and ξ normal to M . Moreover, h is the second fundamental form, D is the linear connection induced in the normal bundle T ⊥ M , called normal connection and A ξ the shape operator in the direction of ξ that is related with h by (6) h(X, Y ), ξ = A ξ X, Y .
For an n−dimensional submanifold M in E m . The mean curvature vector H is given by A submanifold M is said to be minimal (respectively, totally geodesic) if H ≡ 0 (respectively, h ≡ 0).
Consider an n−dimensional Riemannian manifold M and denote by (g ij ) the local components of its metric. Put G = det(g ij ) and (g ij ) = (g ij ) −1 .
Then the Laplacian ∆ of the metric g can be locally defined by for any function u on M , where x 1 , x 2 , ..., x n are local coordinates [11].
M is said to be of finite type if each component of the position vector x has a finite spectral decomposition [2] (8) If all eigenvalues λ 1 , λ 2 , . . . , λ k are mutually distinct, then the immersion x (or the submanifold M ) is said to be of k−type [2].
Definition 1. Frenet curve of rank r for which κ 1 , κ 2 , . . . , κ r−1 are constant is called (generalized) screw line or helix [6]. Since these curves are trajectories of the 1-parameter group of the Euclidean transformations, so, F. Klein and S. Lie [9] called them W − curves.
A unit speed W −curve of rank 2k has the parametrization form and a unit speed W −curve of rank (2k + 1) has the parametrization form where a 0 , b 0 , a 1 , . . . , a k , b 1 , . . . , b k are constant vectors in E m and µ 1 < µ 2 < · · · < µ k are positive real numbers. So, a W −curve of rank 1 is a straight line, a W −curve of rank 2 is a circle and a W −curve of rank 3 is a right circular helix [6].
A W −curve is closed if and only if its rank is even and all µ i are rational multiples of a real number. Therefore, up to rigid motions of a Euclidean space, a closed W −curve of rank 2k and of length 2πr in E 2k has an arc length parameterization of the form: where t 1 < · · · < t k are positive integers [8].

Curves of restricted type
Definition 2. A submanifold M n in E m is said to be restricted type if the shape operator A H is the restriction of a fixed endomorphism A of E m on the tangent space of M n at every point of M n , i.e.

(14)
A H X = (AX) T for any vector X, tangent to M n , where (AX) T denotes the tangential component of AX [7].
Proposition 1. Every submanifold M n in E m whose position vector field satisfies ∆x =Ãx + B, where ∆ is the Laplacian of M n ,Ã ∈ R m×m and B ∈ E m , is of restricted type. The endomorphism A is given by 1 nÃ in this case [7].
Let γ be a regular curve in E m . The Laplacian of γ can be expressed as . By the using of (1) and (15), where H is the mean curvature of γ.
Proposition 2. Let γ be a curve in E m . If γ has the equation such that B is a fixed vector in E m and A a symmetric matrix in R m×m , then γ is of restricted type.
Proof. From Preposition 1 we have the equation where A is a symmetric matrix in R m×m .
Thus S 1 (a) ⊂ E 2 is of restricted type.

Curves of Restricted Type in Euclidean Spaces
Example 2. A helix which is given by the parametrization γ(t) = (r cos(ct + d), r sin(ct + d), at + b) is of restricted type. From higher order derivatives of γ we get Thus helix is of restricted type.
Example 3. Every k−type curve which lies fully in E 2k is of restricted type [7].
Example 4. Every 2−type curve in E m is of restricted type [7].
Using (22), (23) and (24) we have Conversely, if f (s) = const. or f (s) = ±s + b then it is easy to show that the curve given with the parametrization (21) is of restricted type.
We also get the following result.
Proposition 4. Let γ be closed W −curve of rank k and of length 2πr in E 2k given by the parametrization (13). Then γ is of restricted type. Proposition 5 ( [7]). Let γ be a planar curve. γ is of restricted type if and only if the curvature κ of γ satisfies the following differential equation where the derivatives are taken with respect to the arc length parameter.