Abstract
In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curve , then has constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves.
1. Introduction
It is known that a Smarandache geometry is a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied, if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes. Smarandache curves are the objects of Smarandache geometry. By definition, if the position vector of a curve is composed by the Frenet frame’s vectors of another curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–7]. The curves lying on a pseudosphere in Minkowski 3-space are characterized in [8].
In this paper we define nonnull and null pseudospherical Smarandache curves according to the Sabban frame of a spacelike curve lying on pseudosphere in Minkowski 3-space. We obtain the geodesic curvature and the expressions for the Sabban frame’s vectors of spacelike and timelike pseudospherical Smarandache curves. We also prove that if the pseudospherical null straight lines are the Smarandache curves of a spacelike pseudospherical curve , then has nonzero constant geodesic curvature. Finally, we give some examples of pseudospherical Smarandache curves in Minkowski 3-space.
2. Basic Concepts
The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by where is a rectangular coordinate system of . Since is an indefinite metric, recall that a nonzero vector can have one of three Lorentzian causal characters: it can be spacelike if , timelike if , and null (lightlike) if . In particular, the norm (length) of a vector is given by and two vectors and are said to be orthogonal if . Next, recall that an arbitrary curve in can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors are, respectively, spacelike, timelike, or null (lightlike) for all [9]. A spacelike or a timelike curve is parameterized by arclength parameter if or , respectively. For any two vectors and in the space , the pseudovector product of and is defined by
Lemma 1. Let , , and be vectors in . Then,(i), (ii), (iii), where is the pseudovector product in .
Lemma 2. In the Minkowski 3-space , the following properties are satisfied [9]:(i)two timelike vectors are never orthogonal;(ii)two null vectors are orthogonal if and only if they are linearly dependent;(iii)timelike vector is never orthogonal to a null vector.
The pseudosphere with center at the origin and of radius in the Minkowski 3-space is a quadric defined by
Let be a curve lying fully in pseudosphere in . Then its position vector is a spacelike, which means that the tangent vector can be a spacelike, a timelike, or a null. Depending on the causal character of , we distinguish the following three cases.
Case 1 ( is a unit spacelike vector). Then we have orthonormal Sabban frame along the curve , where is the unit timelike vector. The corresponding Frenet formulae of , according to the Sabban frame, read where is the geodesic curvature of and is the arclength parameter of . In particular, the following relations hold:
Case 2 ( is a unit timelike vector). Hence, we have orthonormal Sabban frame along the curve , where is the unit spacelike vector. The corresponding Frenet formulae of , according to the Sabban frame, read where is the geodesic curvature of and is the arclength parameter of . In particular, the following relations hold:
Case 3 ( is a null vector). It is known that the only null curves lying on pseudosphere are the null straight lines, which are the null geodesics.
3. Spacelike and Timelike Pseudospherical Smarandache Curves in Minkowski 3-Space
In this section, we consider spacelike pseudospherical curve and define its spacelike and timelike pseudospherical Smarandache curves according to the Sabban frame of in Mikowski 3-space. Let be a unit speed spacelike curve with the Sabban frame , lying fully on pseudosphere in . Denote by arbitrary nonnull curve lying on pseudosphere, where is the arclength parameter of . Then we have the following definitions of special pseudospherical Smarandache curves of .
Definition 3. Let be a unit speed spacelike curve lying fully on pseudosphere . The nonnull pseudospherical -Smarandache curve of is defined by where is arclength parameter of , , and .
Definition 4. Let be a unit speed spacelike curve lying fully on pseudosphere . The nonnull pseudospherical -Smarandache curve of is defined by where is arclength parameter of , , and .
Definition 5. Let be a unit speed spacelike curve lying fully on pseudosphere . The nonnull pseudospherical -Smarandache curve of is defined by where is arclength parameter of , , and .
Definition 6. Let be a unit speed spacelike curve lying fully on pseudosphere . The nonnull pseudospherical -Smarandache curve of is defined by where is arclength parameter of , , and .
Note that if is a timelike pseudospherical curve, the corresponding nonnull pseudospherical Smarandache curves according to the Sabban frame of can be defined in analogous way. In particular, if is a null pseudospherical curve, then it is a null straight line, so the vectors and are linearly dependent. Thus in this case we do not have the orthonormal Sabban frame of .
Next we obtain the Sabban frame and the geodesic curvature of some special spacelike and timelike pseudospherical Smarandache curves of . We consider the following two cases: (i) is a spacelike curve and (ii) is a timelike curve.
Case 4 ( is a spacelike curve). Then, we have the following theorem.
Theorem 7. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a spacelike pseudospherical -Smarandache curve of , then its frame is given by and the corresponding geodesic curvature reads where , , and .
Proof. Differentiating (8) with respect to and using (4) we obtain
and hence
where
Therefore, the unit spacelike tangent vector of the curve is given by
where if for all and if for all .
Differentiating (17) with respect to and using (4) we find
In particular, from (16) and (18) we get
Since and are spacelike vectors, from (8) and (17) we obtain that the unit timelike vector is given by
Consequently, the geodesic curvature of is given by
Theorem 8. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a spacelike pseudospherical -Smarandache curve of , then its frame is given by and the corresponding geodesic curvature reads where , , and for all .
Proof. Differentiating (9) with respect to and using (4) we obtain
and consequently
where
Therefore, the unit spacelike tangent vector of the curve is given by
where for all .
Differentiating (28) with respect to , it follows that
where
From (27) and (29) we get
On the other hand, from (9) and (28) it can be easily seen that the unit timelike vector is given by
Therefore, the geodesic curvature of is given by
Theorem 9. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a spacelike pseudospherical -Smarandache curve of , then its frame is given by and the corresponding geodesic curvature reads where , , and for all .
Proof. Differentiating (10) with respect to and using (4) we obtain
and consequently
where
It follows that the unit spacelike tangent vector of the curve is given by
where for all . Differentiating (40) with respect to , we find
where
From (39) and (41) we get
Equations (10) and (40) imply
Finally, the geodesic curvature of the curve is given by
Theorem 10. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a spacelike pseudospherical -Smarandache curve of , then its frame is given byand the corresponding geodesic curvature reads where , , and for all .
Proof. Differentiating (11) with respect to and by using (4) we find
and thus
where
Therefore, the unit spacelike tangent vector of the curve is given by
where for all .
Differentiating (52) with respect to and using (51), it follows that
where
From (11) and (52) we get
Consequently, the geodesic curvature of reads
Case 5 ( is a timelike curve). Then, we have the following theorem.
Theorem 11. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . Then the timelike pseudospherical -Smarandache curve of does not exist.
Proof. Assume that there exists a timelike pseudospherical -Smarandache curve of . Differentiating (8) with respect to and using (4) we obtain where is the acrlength parameter of . The previous equation implies This means that a timelike vector is collinear with a spacelike vector , which is a contradiction.
In the theorems which follow, in a similar way as in the Case 4, we obtain the Sabban frame and geodesic curvature of a timelike pseudospherical Smarandache curve . We omit the proofs of Theorems 11, 12, and 13, since they are analogous to the proofs of Theorems 8, 9, and 10.
Theorem 12. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a timelike pseudospherical -Smarandache curve of , then its frame is given by and the corresponding geodesic curvature reads where , , and for all .
Theorem 13. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a timelike pseudospherical -Smarandache curve of , then its frame is given by and the corresponding geodesic curvature reads where , , and for all .
Theorem 14. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a timelike pseudospherical -Smarandache curve of , then its frame is given byand the corresponding geodesic curvature reads where , , and for all .
Corollary 15. If is a spacelike geodesic curve on pseudosphere in Minkowski 3-space , then(1)the spacelike and timelike pseudospherical -Smarandache curves are also geodesic on ;(2)the spacelike and timelike pseudospherical and -Smarandache curves have constant geodesic curvatures on ;(3)the spacelike pseudospherical -Smarandache curve has constant geodesic curvature on .
4. Null Pseudospherical Smarandache Curves in Minkowski 3-Space
In this section, we give definitions of null pseudospherical Smarandache curves which are analogous to the definitions of nonnull pseudospherical Smarandache curves of , given in Section 3.
Definition 16. Let and be a unit speed spacelike and a null curve, respectively, lying fully in pseudosphere . The curve is pseudospherical -Smarandache curve of , if it is given by where , and .
Definition 17. Let and be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere . The curve is pseudospherical -Smarandache curve of , if it is given by where , and .
Definition 18. Let and be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere . The curve is pseudospherical -Smarandache curve of , if it is given by where , and .
Definition 19. Let and be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere . The curve is pseudospherical -Smarandache curve of , if it is given by where , and .
Theorem 20. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . Then the null pseudospherical -Smarandache curve of does not exist.
Proof. Assume that there exists null pseudospherical -Smarandache curve of . Differentiating (68) with respect to and using (4) we obtain This means that a null vector is collinear with a spacelike vector , which is a contradiction.
Theorem 21. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a null pseudospherical -Smarandache curve of , then has constant geodesic curvature given by
Proof. Differentiating (69) with respect to and using (4) we obtain and consequently where , and . The condition implies which proves the theorem.
The next two theorems can be proved in a similar way, so we omit their proofs.
Theorem 22. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a null pseudospherical -Smarandache curve of , then has constant geodesic curvature given by
Theorem 23. Let be a unit speed spacelike curve lying fully in with the Sabban frame and the geodesic curvature . If is a null pseudospherical -Smarandache curve of , then has constant geodesic curvature given by where , , and .
Corollary 24. There are no spacelike pseudospherical geodesic curves whose pseudospherical , , and -Smarandache curves are the null curves.
5. Examples
Example 1. Let be a unit speed spacelike curve lying on pseudosphere in the Minkowski 3-space with parameter equation (see Figure 1) The orthonormal Sabban frame along the curve is given by In particular, the geodesic curvature of the curve has the form
Case 1. Taking and and using (8), we obtain that pseudospherical -Smarandache curve is given by (see Figure 2) It can be easily checked that , which means that is a spacelike curve. According to Theorem 7, its Sabban frame is given by and the corresponding geodesic curvature reads
Case 2. Taking and and using (9) we get that pseudospherical -Smarandache curve is given by (see Figure 3) It can be easily checked that , which means that is a spacelike curve. According to Theorem 8, its frame is given by and the corresponding geodesic curvature reads
Case 3. Taking and , from (10) we find that the pseudospherical -Smarandache curve is given by (see Figure 4) It can be easily checked that , which means that is a spacelike curve. According to Theorem 9, its frame is given by and the corresponding geodesic curvature has the following form:
Case 4. Taking , , and and using (11), we find that the pseudospherical -Smarandache curve has parameter equation (see Figure 5):
It can be easily checked that , which means that is a spacelike curve. By Theorem 10, its frame is given by
while the corresponding geodesic curvature has the following form:
Special spacelike Smarandache curves of and the curve on are shown in Figure 6.
Example 2. Let us consider a unit speed spacelike circle lying on pseudosphere in the Minkowski 3-space with parameter equation (see Figure 7):
The orthonormal Sabban frame along the curve is given by
In particular, the geodesic curvature of reads
Case 1. Taking and and using (9), we obtain that pseudospherical -Smarandache curve is given by (see Figure 8) It can be easily checked that , which means that is a timelike curve. According to Theorem 12, its Sabban frame is given by and the corresponding geodesic curvature reads
Case 2. Taking and and using (10), we obtain that pseudospherical -Smarandache curve is given by (see Figure 9) It can be easily checked that , which means that is a timelike curve. According to Theorem 13, its Sabban frame is given by and the corresponding geodesic curvature reads
Case 3. Taking , , and and using (11), we obtain that pseudospherical -Smarandache curve is given by (see Figure 10)
It can be easily checked that , which means that is a timelike curve. According to Theorem 14, its Sabban frame is given by
and the corresponding geodesic curvature reads
Special timelike Smarandache curves of and the curve on are shown in Figure 11.
Example 3. Let us consider a unit speed spacelike circle lying on pseudosphere in the Minkowski 3-space with parameter (94). Then its geodesic curvature is given by (96).
Case 1. If is null pseudospherical -Smarandache curve of , then according to Theorem 21 the curve has nonzero constant geodesic curvature given by The last relation together with (96) implies . By Definition 17 there holds and therefore we can take . Finally, by using (69) we obtain that pseudospherical -Smarandache curve of is given by (see Figure 12) It can be easily checked that , which means that is a null straight line.
Case 2. If is null pseudospherical -Smarandache curve of , then according to Theorem 22 the curve has nonzero constant geodesic curvature given by The last relation together with (96) implies . By Definition 18 there holds and therefore we can take . Finally, by using (70) we obtain that pseudospherical -Smarandache curve of is given by (see Figure 13) It can be easily checked that , which means that is a null straight line.
Case 3. If is null pseudospherical -Smarandache curve of , then according to Theorem 23 the curve has nonzero constant geodesic curvature given by The last relation together with (96) implies , . By Definition 19 there holds and therefore we can take . Finally, by using (71) we obtain that pseudospherical -Smarandache curve of is given by (see Figure 14) It can be easily checked that , which means that is a null straight line.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.