Definition 2.1 Let α be a non-null curve of osculating order 3. The harmonic functions
defined by
are called harmonic curvatures of α, where , are Frenet curvatures of α which are not necessarily constant.
Definition 2.2 Let α be a time-like curve in with . being a constant unit vector field, if
then α is called a general helix (inclined curves) in , φ is called a slope angle and the space is called a slope axis [5].
Definition 2.3 Let α be a non-null of osculating order 3. Then α is called a general helix of rank1 if
holds, where is a real constant.
We have the following results.
Corollary 2.1
-
(i)
If , then α is a straight line.
-
(ii)
If is constant, then α is a general helix of rank1.
Proof By the use of the above definition, we obtain the proof. □
Proposition 2.1 Let α be a curve in of osculating order 3. Then
where is harmonic curvature of α.
Proof By using the Frenet formulas and the definitions of harmonic curvatures, we get the result. □
Now, we will give the relation between harmonic curvature and a ccr-curve in .
Definition 2.4 A curve is said to have constant curvature ratios (that is to say, it is a ccr-curve) if all the quotients are constant, where .
Corollary 2.2 For , the ccr-curve is .
Proof The proof can be easily seen by using the definitions of harmonic curvature and ccr-curve. □
Corollary 2.3 Let be a ccr-curve. If , c is a constant, then .
Proof The proof is obvious. □