On Pseudospherical Smarandache Curves in Minkowski 3-Space

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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 404521, 14 pages http://dx.doi.org/10.1155/2014/404521

Research Article On Pseudospherical Smarandache Curves in Minkowski 3-Space Esra Betul Koc Ozturk,1,2 Ufuk Ozturk,1,2 Kazim Ilarslan,3 and Emilija NeĹĄoviT4 1

Department of Mathematics, Faculty of Sciences, University of C ¸ ankiri Karatekin, 18100 C ¸ ankiri, Turkey School of Mathematics & Statistical Sciences, Room PSA442, Arizona State University, Tempe, AZ 85287-1804, USA 3 Department of Mathematics, Faculty of Sciences and Arts, University of Kirikkale, 71450 Kirikkale, Turkey 4 Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia 2

Correspondence should be addressed to Ufuk Ozturk; uuzturk@asu.edu Received 3 June 2013; Accepted 21 November 2013; Published 24 February 2014 Academic Editor: Laurent Gosse Copyright Š 2014 Esra Betul Koc Ozturk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 đ?›ź, then đ?›ź 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 đ?›˝ is composed by the Frenet frame’s vectors of another curve đ?›ź, then the curve đ?›˝ 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 đ?‘†12 in Minkowski 3-space đ??¸13 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 đ?›ź, then đ?›ź has nonzero constant geodesic curvature. Finally, we give some examples

of pseudospherical Smarandache curves in Minkowski 3space.

2. Basic Concepts The Minkowski 3-space R31 is the Euclidean 3-space R3 provided with the standard flat metric given by â&#x;¨â‹…, â‹…â&#x;Š = − đ?‘‘đ?‘Ľ12 + đ?‘‘đ?‘Ľ22 + đ?‘‘đ?‘Ľ32 ,

(1)

where (đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 ) is a rectangular coordinate system of R31 . Since đ?‘” is an indefinite metric, recall that a nonzero vector đ?‘Ľâƒ— ∈ R31 can have one of three Lorentzian causal characters: it can be spacelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— > 0, timelike if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— < 0, and null (lightlike) if â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâƒ— = 0. In particular, the norm (length) of a ⃗ and two vectors đ?‘Ľâƒ— vector đ?‘Ľâƒ— ∈ R31 is given by ‖đ?‘Ľâ€–⃗ = √|â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Š| and đ?‘Śâƒ— are said to be orthogonal if â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— = 0. Next, recall that an arbitrary curve đ?›ź = đ?›ź(đ?‘ ) in E31 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors đ?›źó¸€ (đ?‘ ) are, respectively, spacelike, timelike, or null (lightlike) for all đ?‘ ∈ đ??ź [9]. A spacelike or a timelike curve đ?›ź is parameterized by arclength parameter đ?‘ if â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = 1 or â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = −1, respectively. For any two vectors đ?‘Ľâƒ— = (đ?‘Ľ1 , đ?‘Ľ2, đ?‘Ľ3 ) and

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đ?‘Śâƒ— = (đ?‘Ś1 , đ?‘Ś2 , đ?‘Ś3 ) in the space R31 , the pseudovector product of đ?‘Ľâƒ— and đ?‘Śâƒ— is defined by óľ„¨óľ„¨âˆ’ đ?‘’ ⃗ đ?‘’ ⃗ đ?‘’ ⃗ 󾄨󾄨 󾄨󾄨 1 2 3 󾄨󾄨 󾄨 󾄨 đ?‘Ľâƒ— Ă— đ?‘Śâƒ— = 󾄨󾄨󾄨 đ?‘Ľ1 đ?‘Ľ2 đ?‘Ľ3 󾄨󾄨󾄨 󾄨 󾄨󾄨 (2) 󾄨󾄨 đ?‘Ś1 đ?‘Ś2 đ?‘Ś3 󾄨󾄨󾄨 = (− đ?‘Ľ2 đ?‘Ś3 + đ?‘Ľ3 đ?‘Ś2 , đ?‘Ľ3 đ?‘Ś1 − đ?‘Ľ1 đ?‘Ś3 , đ?‘Ľ1 đ?‘Ś2 − đ?‘Ľ2 đ?‘Ś1 ) . Lemma 1. Let đ?‘Ľ,⃗ đ?‘Ś,⃗ and đ?‘§âƒ— be vectors in R31 . Then,

where đ?‘˜đ?‘” (đ?‘ ) = det(đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?‘‡ó¸€ (đ?‘ )) is the geodesic curvature of đ?›ź and đ?‘ is the arclength parameter of đ?›ź. In particular, the following relations hold: đ?›ź Ă— đ?‘‡ = đ?œ‰,

đ?‘‡ Ă— đ?œ‰ = đ?›ź,

đ?œ‰ Ă— đ?›ź = −đ?‘‡.

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Case 3 (� is a null vector). It is known that the only null curves lying on pseudosphere �12 are the null straight lines, which are the null geodesics.

⃗ (i) â&#x;¨đ?‘Ľâƒ— Ă— đ?‘Ś,⃗ đ?‘§â&#x;Šâƒ— = det(đ?‘Ľ,⃗ đ?‘Ś,⃗ đ?‘§), (ii) đ?‘Ľâƒ— Ă— (đ?‘Śâƒ— Ă— đ?‘§)⃗ = −â&#x;¨đ?‘Ľ,⃗ đ?‘§â&#x;Šâƒ— đ?‘Śâƒ— + â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— đ?‘§,⃗

3. Spacelike and Timelike Pseudospherical Smarandache Curves in Minkowski 3-Space

⃗ đ?‘Ś,⃗ đ?‘Śâ&#x;Šâƒ— + â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— 2 , (iii) â&#x;¨đ?‘Ľâƒ— Ă— đ?‘Ś,⃗ đ?‘Ľâƒ— Ă— đ?‘Śâ&#x;Šâƒ— = −â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâ&#x;¨

In this section, we consider spacelike pseudospherical curve đ?›ź and define its spacelike and timelike pseudospherical Smarandache curves according to the Sabban frame of đ?›ź in Mikowski 3-space. Let đ?›ź = đ?›ź(đ?‘ ) be a unit speed spacelike curve with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰}, lying fully on pseudosphere đ?‘†12 in R31 . Denote by đ?›˝ = đ?›˝(đ?‘ ∗ ) arbitrary nonnull curve lying on pseudosphere, where đ?‘ ∗ is the arclength parameter of đ?›˝. Then we have the following definitions of special pseudospherical Smarandache curves of đ?›ź.

where đ?‘Ľ is the pseudovector product in

R31 .

Lemma 2. In the Minkowski 3-space R31 , 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 đ?‘&#x; = 1 in the Minkowski 3-space R31 is a quadric defined by đ?‘†12 = {đ?‘Ľâƒ— ∈ R31 | −đ?‘Ľ12 + đ?‘Ľ22 + đ?‘Ľ32 = 1} .

(3)

Let đ?›ź : đ??ź ⊂ R → đ?‘†12 be a curve lying fully in pseudosphere đ?‘†12 in R31 . Then its position vector đ?›ź is a spacelike, which means that the tangent vector đ?‘‡ = đ?›źó¸€ can be a spacelike, a timelike, or a null. Depending on the causal character of đ?‘‡, we distinguish the following three cases.

Definition 3. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully on pseudosphere đ?‘†12 . The nonnull pseudospherical đ?›źđ?œ‰-Smarandache curve đ?›˝ of đ?›ź is defined by đ?›˝ (đ?‘ ∗ (đ?‘ )) =

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √2

(8)

where đ?‘ ∗ is arclength parameter of đ?›˝, đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 − đ?‘?2 = 2. Definition 4. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully on pseudosphere đ?‘†12 . The nonnull pseudospherical đ?›źđ?‘‡-Smarandache curve đ?›˝ of đ?›ź is defined by

Case 1 (đ?‘‡ is a unit spacelike vector). Then we have orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?œ‰(đ?‘ )} along the curve đ?›ź, where đ?œ‰(đ?‘ ) = −đ?›ź(đ?‘ )Ă—đ?‘‡(đ?‘ ) is the unit timelike vector. The corresponding Frenet formulae of đ?›ź, according to the Sabban frame, read

where đ?‘ ∗ is arclength parameter of đ?›˝, đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 + đ?‘?2 = 2.

0 1 0 đ?›źó¸€ đ?›ź [đ?‘‡ó¸€ ] = [−1 0 −đ?‘˜đ?‘” (đ?‘ )] [đ?‘‡] , ó¸€ 0 ] [đ?œ‰] [ đ?œ‰ ] [ 0 −đ?‘˜đ?‘” (đ?‘ )

Definition 5. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully on pseudosphere đ?‘†12 . The nonnull pseudospherical đ?‘‡đ?œ‰-Smarandache curve đ?›˝ of đ?›ź is defined by

(4)

where đ?‘˜đ?‘” (đ?‘ ) = det(đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?‘‡ó¸€ (đ?‘ )) is the geodesic curvature of đ?›ź and đ?‘ is the arclength parameter of đ?›ź. In particular, the following relations hold: đ?›ź Ă— đ?‘‡ = −đ?œ‰,

đ?‘‡ Ă— đ?œ‰ = đ?›ź,

đ?œ‰ Ă— đ?›ź = đ?‘‡.

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Case 2 (đ?‘‡ is a unit timelike vector). Hence, we have orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?œ‰(đ?‘ )} along the curve đ?›ź, where đ?œ‰(đ?‘ ) = đ?›ź(đ?‘ )Ă—đ?‘‡(đ?‘ ) is the unit spacelike vector. The corresponding Frenet formulae of đ?›ź, according to the Sabban frame, read 0 1 0 đ?›źó¸€ đ?›ź [đ?‘‡ó¸€ ] = [1 0 đ?‘˜đ?‘” (đ?‘ )] [đ?‘‡] , ó¸€ [ đ?œ‰ ] [0 đ?‘˜đ?‘” (đ?‘ ) 0 ] [ đ?œ‰ ]

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đ?›˝ (đ?‘ ∗ (đ?‘ )) =

đ?›˝ (đ?‘ ∗ (đ?‘ )) =

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?‘‡ (đ?‘ )) , √2

1 (đ?‘Žđ?‘‡ (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √2

(9)

(10)

where đ?‘ ∗ is arclength parameter of đ?›˝, đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 − đ?‘?2 = 2. Definition 6. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully on pseudosphere đ?‘†12 . The nonnull pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve đ?›˝ of đ?›ź is defined by đ?›˝ (đ?‘ ∗ (đ?‘ )) =

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?‘‡ (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √3

(11)

where đ?‘ ∗ is arclength parameter of đ?›˝, đ?‘Ž, đ?‘?, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 + đ?‘?2 − đ?‘?2 = 3.

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Note that if đ?›ź is a timelike pseudospherical curve, the corresponding nonnull pseudospherical Smarandache curves according to the Sabban frame of đ?›ź can be defined in analogous way. In particular, if đ?›ź is a null pseudospherical curve, then it is a null straight line, so the vectors đ?›źĂ—đ?‘‡ and đ?‘‡ are linearly dependent. Thus in this case we do not have the orthonormal Sabban frame of đ?›ź. Next we obtain the Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } and the geodesic curvature đ?œ…đ?‘”đ?›˝ of some special spacelike and timelike pseudospherical Smarandache curves of đ?›ź. We consider the following two cases: (i) đ?›˝ is a spacelike curve and (ii) đ?›˝ is a timelike curve.

Differentiating (17) with respect to đ?‘ and using (4) we find đ?‘‘đ?‘‡đ?›˝ đ?‘‘đ?‘ ∗ đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

đ?‘? đ?‘Ž [ √2 0 √2 ] đ?›ź đ?›˝ [ ] [đ?‘‡đ?›˝ ] = [ 0 đ?œ– 0 ] [đ?‘‡] , [ ] [ đ?‘? đ?‘Ž ] [đ?œ‰] [ đ?œ‰đ?›˝ ] 0 đ?œ– đ?œ– √2 ] [ √2

√2đ?œ– đ?‘‡đ?›˝ó¸€ = 󾄨󾄨 󾄨 (−đ?›ź − đ?‘˜đ?‘” đ?œ‰) . 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

đ?‘? − đ?‘Žđ?‘˜đ?‘” đ?‘˜đ?‘”đ?›˝ = 󾄨󾄨 󾄨, 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

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where đ?‘Ž, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 − đ?‘?2 = 2, and đ?œ– = Âą1. Proof. Differentiating (8) with respect to đ?‘ and using (4) we obtain đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡, đ?›˝ (đ?‘ ) = ∗ = √2 đ?‘‘đ?‘ đ?‘‘đ?‘ ó¸€

(14)

and hence ��

đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡, = √2 đ?‘‘đ?‘

(15)

where 󾄨󾄨 󾄨 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 đ?‘‘đ?‘ ∗ 󾄨 󾄨. ó¸€ ó¸€ = √â&#x;¨đ?›˝ (đ?‘ ) , đ?›˝ (đ?‘ )â&#x;Š = √2 đ?‘‘đ?‘

(16)

Therefore, the unit spacelike tangent vector of the curve đ?›˝ is given by đ?‘‡đ?›˝ = đ?œ–đ?‘‡,

đ?œ‰đ?›˝ = −đ?›˝ Ă— đ?‘‡đ?›˝ =đ?œ–

(20)

đ?‘Ž đ?‘? đ?›ź + đ?œ– đ?œ‰. √2 √2

Consequently, the geodesic curvature đ?‘˜đ?‘”đ?›˝ of đ?›˝ is given by đ?‘˜đ?‘”đ?›˝ = det (đ?›˝, đ?‘‡đ?›˝ , đ?‘‡đ?›˝ó¸€ ) đ?‘? − đ?‘Žđ?‘˜đ?‘” = 󾄨󾄨 󾄨. 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

(21)

Theorem 8. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a spacelike pseudospherical đ?›źđ?‘‡-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by đ?›˝ [đ?‘‡đ?›˝ ] [ đ?œ‰đ?›˝ ] đ?‘? đ?‘Ž 0 [ ] √ √ 2 2 [ ] [ ] −đ?‘?đ?‘˜đ?‘” [ ] −đ?‘? đ?‘Ž [ ] [ ] 2 2 2 ] [ = [ √ 2 − (đ?‘?đ?‘˜ ) √ 2 − (đ?‘?đ?‘˜đ?‘” ) √ 2 − (đ?‘?đ?‘˜đ?‘” ) ] đ?‘” [ ] [ ] đ?‘?2 đ?‘˜đ?‘” −đ?‘Žđ?‘?đ?‘˜đ?‘” [ ] 2 [ ] [ 2 2 2] √ 4 − 2(đ?‘?đ?‘˜đ?‘” ) √ 4 − 2(đ?‘?đ?‘˜đ?‘” ) √ 4 − 2(đ?‘?đ?‘˜đ?‘” ) [ ] đ?›ź Ă— [đ?‘‡] , [đ?œ‰]

(22)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads

(17)

where đ?œ– = +1 if đ?‘Ž − đ?‘?đ?‘˜đ?‘” > 0 for all đ?‘ and đ?œ– = −1 if đ?‘Ž − đ?‘?đ?‘˜đ?‘” < 0 for all đ?‘ .

(19)

Since đ?›˝ and đ?‘‡đ?›˝ are spacelike vectors, from (8) and (17) we obtain that the unit timelike vector đ?œ‰đ?›˝ is given by

(12)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads

(18)

In particular, from (16) and (18) we get

Case 4 (đ?›˝ is a spacelike curve). Then, we have the following theorem. Theorem 7. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a spacelike pseudospherical đ?›źđ?œ‰-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by

= đ?œ– (−đ?›ź − đ?‘˜đ?‘” đ?œ‰) .

đ?‘˜đ?‘”đ?›˝

=

−đ?‘˜đ?‘” đ?‘?2 đ?œ€1 + đ?‘Žđ?‘˜đ?‘” đ?‘?đ?œ€2 + 2đ?œ€3 5/2

(2 − đ?‘?2 đ?‘˜đ?‘”2 )

,

(23)

4

Journal of Applied Mathematics Therefore, the geodesic curvature đ?‘˜đ?‘”đ?›˝ of đ?›˝ is given by

where đ?œ€1 = − đ?‘?3 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + đ?‘Žđ?‘?2 đ?‘˜đ?‘”2 − 2đ?‘Ž, đ?œ€2 =

đ?‘Žđ?‘?2 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” 2đ?‘?đ?‘˜đ?‘”ó¸€

đ?œ€3 = − 2

−

+

đ?‘?3 đ?‘˜đ?‘”4

+

đ?‘Žđ?‘?2 đ?‘˜đ?‘”3

2

đ?‘Ž, đ?‘? ∈ đ?‘…0 , đ?‘Ž + đ?‘? = 2, and

đ?‘?2 đ?‘˜đ?‘”2

2đ?‘?đ?‘˜đ?‘”2

+

đ?‘?3 đ?‘˜đ?‘”2

− 2đ?‘?,

(24)

=

− 2đ?‘Žđ?‘˜đ?‘” ,

đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ 1 (−đ?‘?đ?›ź + đ?‘Žđ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , = √2 đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

(25)

and consequently ��

đ?‘‘đ?‘ ∗ 1 (−đ?‘?đ?›ź + đ?‘Žđ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , = √2 đ?‘‘đ?‘

(26)

where 2

đ?‘‘đ?‘ ∗ √ 2 − (đ?‘?đ?‘˜đ?‘” ) = . đ?‘‘đ?‘ 2

(27)

Therefore, the unit spacelike tangent vector of the curve � is given by �� =

1 2

√ 2 − (đ?‘?đ?‘˜đ?‘” )

(−đ?‘?đ?›ź + đ?‘Žđ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) ,

(28)

đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

1

=

2 3/2

(2 − (đ?‘?đ?‘˜đ?‘” ) )

(2 − đ?‘?2 đ?‘˜đ?‘”2 )

5/2

.

(33)

Theorem 9. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a spacelike pseudospherical đ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by đ?‘? đ?‘Ž 0 [ ] √2 √2 [ ] [ ] [ ] [ ] −đ?‘?đ?‘˜ −đ?‘Žđ?‘˜ −đ?‘Ž đ?‘” đ?‘” đ?›˝ [ ] đ?›ź [ ] [đ?‘‡] [đ?‘‡đ?›˝ ] = [ đ?‘Ž2 − 2đ?‘˜2 2 2 2 2 , √ √đ?‘Ž − 2đ?‘˜đ?‘” √đ?‘Ž − 2đ?‘˜đ?‘” ] đ?‘” [ ] đ?œ‰ đ?œ‰ [ ] [ đ?›˝] [ ][ ] [ ] 2 2đ?‘˜ [ ] đ?‘Žđ?‘? đ?‘Ž đ?‘” [ ] 2 2 2 2 2 2 2đ?‘Ž − 4đ?‘˜đ?‘” √2đ?‘Ž − 4đ?‘˜đ?‘” √2đ?‘Ž − 4đ?‘˜đ?‘” √ [ ] (34) and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads

where 2 − (đ?‘?đ?‘˜đ?‘” )2 > 0 for all đ?‘ . Differentiating (28) with respect to đ?‘ , it follows that đ?‘‘đ?‘‡đ?›˝ đ?‘‘đ?‘ ∗

−đ?‘˜đ?‘” đ?‘?2 đ?œ€1 + đ?‘Žđ?‘˜đ?‘” đ?‘?đ?œ€2 + 2đ?œ€3

− 2 < 0 for all đ?‘ .

Proof. Differentiating (9) with respect to đ?‘ and using (4) we obtain đ?›˝ó¸€ (đ?‘ ) =

đ?‘˜đ?‘”đ?›˝ = det (đ?›˝, đ?‘‡đ?›˝ , đ?‘‡đ?›˝ó¸€ )

(đ?œ€1 đ?›ź + đ?œ€2 đ?‘‡ + đ?œ€3 đ?œ‰) ,

đ?‘˜đ?‘”đ?›˝ =

(29)

−2đ?‘˜đ?‘” đ?œ€1 − đ?‘?đ?‘Žđ?œ€2 + đ?œ€3 đ?‘Ž2 (đ?‘Ž2 − 2đ?‘˜đ?‘”2 )

5/2

,

(35)

where

where đ?œ€1 = −

đ?‘?3 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘”

+

đ?‘Žđ?‘?2 đ?‘˜đ?‘”2

− 2đ?‘Ž,

đ?œ€1 = − 2đ?‘Žđ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” − 2đ?‘?đ?‘˜đ?‘”3 + đ?‘Ž2 đ?‘?đ?‘˜đ?‘” ,

đ?œ€2 = đ?‘Žđ?‘?2 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” − đ?‘?3 đ?‘˜đ?‘”4 + 2đ?‘?đ?‘˜đ?‘”2 + đ?‘?3 đ?‘˜đ?‘”2 − 2đ?‘?, đ?œ€3 = −

2đ?‘?đ?‘˜đ?‘”ó¸€

+

đ?‘Žđ?‘?2 đ?‘˜đ?‘”3

(30)

− 2đ?‘Žđ?‘˜đ?‘” .

đ?œ€2 = − 2đ?‘Žđ?‘˜đ?‘”4 + (đ?‘Ž3 + 2đ?‘Ž) đ?‘˜đ?‘”2 − đ?‘Ž3 − đ?‘Ž2 đ?‘?đ?‘˜đ?‘”ó¸€ ,

(36)

đ?œ€3 = − đ?‘Ž3 đ?‘˜đ?‘”ó¸€ âˆ’ 2đ?‘?đ?‘˜đ?‘”4 + đ?‘Ž2 đ?‘?đ?‘˜đ?‘”2 ,

From (27) and (29) we get đ?‘‡đ?›˝ó¸€ =

1 2 2

(2 − (đ?‘?đ?‘˜đ?‘” ) )

(đ?œ€1 đ?›ź + đ?œ€2 đ?‘‡ + đ?œ€3 đ?œ‰) .

(31)

On the other hand, from (9) and (28) it can be easily seen that the unit timelike vector đ?œ‰đ?›˝ is given by đ?œ‰đ?›˝ = −đ?›˝ Ă— đ?‘‡đ?›˝ =

đ?‘?2 đ?‘˜đ?‘” 2

√ 4 − 2(đ?‘?đ?‘˜đ?‘” )

đ?›źâˆ’

đ?‘Žđ?‘?đ?‘˜đ?‘” 2

√ 4 − 2(đ?‘?đ?‘˜đ?‘” )

�+

2 2

đ?œ‰.

đ?‘Ž, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 − đ?‘?2 = 2, and đ?‘Ž2 − 2đ?‘˜đ?‘”2 > 0 for all đ?‘ . Proof. Differentiating (10) with respect to đ?‘ and using (4) we obtain đ?›˝ó¸€ (đ?‘ ) =

đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ 1 (−đ?‘Žđ?›ź − đ?‘?đ?‘˜đ?‘” đ?‘‡ − đ?‘Žđ?‘˜đ?‘” đ?œ‰) , = √2 đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

and consequently

√ 4 − 2(đ?‘?đ?‘˜đ?‘” )

(32)

(37)

��

đ?‘‘đ?‘ ∗ 1 (−đ?‘Žđ?›ź − đ?‘?đ?‘˜đ?‘” đ?‘‡ − đ?‘Žđ?‘˜đ?‘” đ?œ‰) , = √2 đ?‘‘đ?‘

(38)

Journal of Applied Mathematics

5

where

From (39) and (41) we get 2 2 𝑑𝑠∗ √ 𝑎 − 2𝑘𝑔 = . 𝑑𝑠 2

It follows that the unit spacelike tangent vector of the curve 𝛽 is given by 1

𝑇𝛽 =

(−𝑎𝛼 − 𝑏𝑘𝑔 𝑇 − 𝑎𝑘𝑔 𝜉) ,

√𝑎2 − 2𝑘𝑔2

(40)

where 𝑎2 − 2𝑘𝑔2 > 0 for all 𝑠. Differentiating (40) with respect to 𝑠, we find 𝑑𝑇𝛽 𝑑𝑠

∗

𝑑𝑠∗ 𝑑𝑠

=

1 (𝑎2

−

2𝑘𝑔2 )

3/2

𝑇𝛽󸀠 =

(39)

(𝜀1 𝛼 + 𝜀2 𝑇 + 𝜀3 𝜉) ,

(41)

1 (𝑎2

− 2𝑘𝑔2 )

2

(𝜀1 𝛼 + 𝜀2 𝑇 + 𝜀3 𝜉) .

(43)

Equations (10) and (40) imply 𝜉𝛽 = −𝛽 × 𝑇𝛽 =

2𝑘𝑔 √2𝑎2 − 4𝑘𝑔2

𝛼+

𝑎𝑏 √2𝑎2 − 4𝑘𝑔2

𝑇+

𝑎2 √2𝑎2 − 4𝑘𝑔2

𝜉.

(44)

Finally, the geodesic curvature 𝑘𝑔𝛽 of the curve 𝛽 is given by 𝑘𝑔𝛽 = det (𝛽, 𝑇𝛽 , 𝑇𝛽󸀠 ) =

−2𝑘𝑔 𝜀1 − 𝑏𝑎𝜀2 + 𝜀3 𝑎2 5/2

(𝑎2 − 2𝑘𝑔2 )

.

(45)

where 𝜀1 = − 2𝑎𝑘𝑔󸀠 𝑘𝑔 − 2𝑏𝑘𝑔3 + 𝑎2 𝑏𝑘𝑔 , 𝜀2 = − 2𝑎𝑘𝑔4 + (𝑎3 + 2𝑎) 𝑘𝑔2 − 𝑎3 − 𝑎2 𝑏𝑘𝑔󸀠 ,

(42)

𝜀3 = − 𝑎3 𝑘𝑔󸀠 − 2𝑏𝑘𝑔4 + 𝑎2 𝑏𝑘𝑔2 .

Theorem 10. Let 𝛼 : 𝐼 ⊂ 𝑅 󳨃→ 𝑆12 be a unit speed spacelike curve lying fully in 𝑆12 with the Sabban frame {𝛼, 𝑇, 𝜉} and the geodesic curvature 𝑘𝑔 . If 𝛽 : 𝐼 ⊂ 𝑅 󳨃→ 𝑆12 is a spacelike pseudospherical 𝛼𝑇𝜉-Smarandache curve of 𝛼, then its frame {𝛽, 𝑇𝛽 , 𝜉𝛽 } is given by

𝑏 𝑐 𝑎 ] [ √ √ √ 3 3 3 ] [ ] [ ] [ −𝑏𝑘𝑔 ] [ −𝑏 𝑎 − 𝑐𝑘 ] [ 𝛽 ] 𝛼 [ 2 2 2 [ [𝑇𝛽 ] = [ √ (𝑎 − 𝑐𝑘𝑔 ) + 𝑏2 (1 − 𝑘2 ) √ (𝑎 − 𝑐𝑘𝑔 ) + 𝑏2 (1 − 𝑘𝑔2 ) √ (𝑎 − 𝑐𝑘𝑔 ) + 𝑏2 (1 − 𝑘𝑔2 ) ] ] [𝑇] 𝑔 ] 𝜉 [ 𝜉 ][ ] [ 𝛽] [ ] [ ] [ 2 2 𝑎 (𝑎 − 𝑐𝑘𝑔 ) + 𝑏 𝑐 (𝑎 − 𝑐𝑘𝑔 ) + 𝑏 𝑘𝑔 ] [ −𝑎𝑏𝑘𝑔 + 𝑏𝑐 ] [ ] [ 2 2 2 √ 3(𝑎 − 𝑐𝑘𝑔 ) + 3𝑏2 (1 − 𝑘𝑔2 ) √ 3(𝑎 − 𝑐𝑘𝑔 ) + 3𝑏2 (1 − 𝑘𝑔2 ) √ 3(𝑎 − 𝑐𝑘𝑔 ) + 3𝑏2 (1 − 𝑘𝑔2 ) ] [ 𝜀2 = (𝑎2 − 3) 𝑐𝑘𝑔󸀠 𝑘𝑔2 + (𝑎𝑏2 − 2𝑎𝑐2 ) 𝑘𝑔󸀠 𝑘𝑔 + 𝑎2 𝑐𝑘𝑔󸀠

and the corresponding geodesic curvature 𝑘𝑔𝛽 reads

+ 𝑏 (𝑎2 − 3) 𝑘𝑔4 − 2𝑎𝑏𝑐𝑘𝑔3 + (𝑏3 + 3𝑏) 𝑘𝑔2

𝑘𝑔𝛽 = (((𝑎2 − 3) 𝑘𝑔 − 𝑎𝑐) 𝜀1 + (𝑎𝑏𝑘𝑔 − 𝑏𝑐) 𝜀2

+ 2𝑎𝑏𝑐𝑘𝑔 − 𝑏 (𝑐2 + 3) ,

2

+ (−𝑎𝑐𝑘𝑔 + 𝑐 + 3) 𝜀3 )

(47)

2

× (((𝑎 − 𝑐𝑘𝑔 ) + 𝑏2 (1 − 𝑘𝑔2 ))

(46)

5/2 −1

) ,

𝜀3 = 2𝑏 (𝑎2 − 3) 𝑘𝑔󸀠 𝑘𝑔2 − 3𝑎𝑏𝑐𝑘𝑔󸀠 𝑘𝑔 + 𝑏 (𝑐2 + 3) 𝑘𝑔󸀠 + 𝑐 (𝑎2 − 3) 𝑘𝑔4 + (𝑎𝑏2 − 3𝑎𝑐2 ) 𝑘𝑔3 + (3𝑎2 𝑐 + 𝑏2 𝑐) 𝑘𝑔2 − 𝑎 (𝑐2 + 3) 𝑘𝑔 ,

where

(48)

𝜀1 = (𝑎2 − 3) 𝑏𝑘𝑔󸀠 𝑘𝑔 − 𝑎𝑏𝑐𝑘𝑔󸀠 + (𝑎2 − 3) 𝑐𝑘𝑔3 2

2

+ (𝑎𝑏 − 3𝑎𝑐

) 𝑘𝑔2

2

2

2

+ (3𝑎 𝑐 + 𝑏 𝑐) 𝑘𝑔 − 𝑎 (𝑐 + 3) ,

𝑎, 𝑏, 𝑐 ∈ 𝑅0 , 𝑎2 + 𝑏2 − 𝑐2 = 3, and (𝑎 − 𝑐𝑘𝑔 )2 + 𝑏2 (1 − 𝑘𝑔2 ) > 0 for all 𝑠.

6

Journal of Applied Mathematics

Proof. Differentiating (11) with respect to đ?‘ and by using (4) we find đ?›˝ó¸€ (đ?‘ ) =

From (11) and (52) we get đ?œ‰đ?›˝ = −đ?›˝ Ă— đ?‘‡đ?›˝

∗

đ?‘‘đ?›˝ đ?‘‘đ?‘ 1 = (−đ?‘?đ?›ź + (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , (49) ∗ √3 đ?‘‘đ?‘ đ?‘‘đ?‘

đ?‘? (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + đ?‘?2 đ?‘˜đ?‘”

=

�

2

√ 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + 3đ?‘?2 (1 − đ?‘˜đ?‘”2 ) and thus đ?‘‘đ?‘ 1 đ?‘‡đ?›˝ (−đ?‘?đ?›ź + (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , = √3 đ?‘‘đ?‘

−đ?‘Žđ?‘?đ?‘˜đ?‘” + đ?‘?đ?‘?

+

∗

(55)

√ 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + 3đ?‘?2 (1 − đ?‘˜đ?‘”2 )

(50)

đ?‘Ž (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + đ?‘?2

+

where

�

2

đ?œ‰.

2

√ 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + 3đ?‘?2 (1 − đ?‘˜đ?‘”2 ) 2

2 2 đ?‘‘đ?‘ ∗ √ (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + đ?‘? (1 − đ?‘˜đ?‘” ) = . đ?‘‘đ?‘ 3

(51)

Therefore, the unit spacelike tangent vector of the curve � is given by �� =

Consequently, the geodesic curvature đ?‘˜đ?‘”đ?›˝ of đ?›˝ reads đ?‘˜đ?‘”đ?›˝ = det (đ?›˝, đ?‘‡đ?›˝ , đ?‘‡đ?›˝ó¸€ ) = (((đ?‘Ž2 − 3) đ?‘˜đ?‘” − đ?‘Žđ?‘?) đ?œ€1 + (đ?‘Žđ?‘?đ?‘˜đ?‘” − đ?‘?đ?‘?) đ?œ€2 (56)

+ (−đ?‘Žđ?‘?đ?‘˜đ?‘” + đ?‘?2 + 3) đ?œ€3 )

1 2

√ (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + đ?‘?2 (1 − đ?‘˜đ?‘”2 )

(52)

2

Ă— (((đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) + đ?‘?2 (1 − đ?‘˜đ?‘”2 ))

5/2 −1

) .

Ă— (−đ?‘?đ?›ź + (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , where (đ?‘Ž − đ?‘?đ?‘˜đ?‘” )2 + đ?‘?2 (1 − đ?‘˜đ?‘”2 ) > 0 for all đ?‘ . Differentiating (52) with respect to đ?‘ and using (51), it follows that đ?‘‡đ?›˝ó¸€ =

1 2

((đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) +

đ?‘?2

(1 −

2 đ?‘˜đ?‘”2 ))

(đ?œ€1 đ?›ź + đ?œ€2 đ?‘‡ + đ?œ€3 đ?œ‰) ,

(53)

Case 5 (đ?›˝ is a timelike curve). Then, we have the following theorem. Theorem 11. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . Then the timelike pseudospherical đ?›źđ?œ‰Smarandache curve of đ?›ź does not exist. Proof. Assume that there exists a timelike pseudospherical đ?›źđ?œ‰-Smarandache curve đ?›˝ of đ?›ź. Differentiating (8) with respect to đ?‘ and using (4) we obtain

where đ?œ€1 = (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” − đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”ó¸€ + (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”3

đ?›˝ó¸€ (đ?‘ ) =

+ (đ?‘Žđ?‘?2 − 3đ?‘Žđ?‘?2 ) đ?‘˜đ?‘”2 + (3đ?‘Ž2 đ?‘? + đ?‘?2 đ?‘?) đ?‘˜đ?‘” − đ?‘Ž (đ?‘?2 + 3) , đ?œ€2 = (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘”2 + (đ?‘Žđ?‘?2 − 2đ?‘Žđ?‘?2 ) đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + đ?‘Ž2 đ?‘?đ?‘˜đ?‘”ó¸€ + đ?‘? (đ?‘Ž2 − 3) đ?‘˜đ?‘”4 − 2đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”3

(57)

where đ?‘ ∗ is the acrlength parameter of đ?›˝. The previous equation implies đ?‘‡đ?›˝

+ (đ?‘?3 + 3đ?‘?) đ?‘˜đ?‘”2 + 2đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘” − đ?‘? (đ?‘?2 + 3) ,

đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡, = √2 đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡. = √2 đ?‘‘đ?‘

(58)

This means that a timelike vector �� is collinear with a spacelike vector �, which is a contradiction.

đ?œ€3 = 2đ?‘? (đ?‘Ž2 − 3) đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘”2 − 3đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + đ?‘? (đ?‘?2 + 3) đ?‘˜đ?‘”ó¸€ + đ?‘? (đ?‘Ž2 − 3) đ?‘˜đ?‘”4 + (đ?‘Žđ?‘?2 − 3đ?‘Žđ?‘?2 ) đ?‘˜đ?‘”3 + (3đ?‘Ž2 đ?‘? + đ?‘?2 đ?‘?) đ?‘˜đ?‘”2 − đ?‘Ž (đ?‘?2 + 3) đ?‘˜đ?‘” . (54)

In the theorems which follow, in a similar way as in the Case 4, we obtain the Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } and geodesic curvature đ?œ…đ?‘”đ?›˝ of a timelike pseudospherical Smarandache curve đ?›˝. We omit the proofs of Theorems 11, 12, and 13, since they are analogous to the proofs of Theorems 8, 9, and 10.

Journal of Applied Mathematics

7

Theorem 12. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a timelike pseudospherical đ?›źđ?‘‡-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by đ?›˝ [đ?‘‡đ?›˝ ] [ đ?œ‰đ?›˝ ] [ ] đ?‘? đ?‘Ž [ ] 0 [ ] √2 √2 [ ] [ ] [ ] −đ?‘?đ?‘˜đ?‘” đ?‘Ž −đ?‘? [ ] [ ] [ ] =[ 2 2 2 ] √ √ √ (đ?‘?đ?‘˜ ) − 2 (đ?‘?đ?‘˜ ) − 2 (đ?‘?đ?‘˜ ) − 2 [ ] đ?‘” đ?‘” đ?‘” [ ] [ ] 2 [ ] −đ?‘? đ?‘˜đ?‘” đ?‘Žđ?‘?đ?‘˜đ?‘” −2 [ ] [ ] [ ] 2 2 2 √ 2(đ?‘?đ?‘˜đ?‘” ) − 4 √ 2(đ?‘?đ?‘˜đ?‘” ) − 4 √ 2(đ?‘?đ?‘˜đ?‘” ) − 4 [ ] đ?›ź Ă— [đ?‘‡] , [đ?œ‰]

(59)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ =

−đ?‘˜đ?‘” đ?‘?2 đ?œ€1 + đ?‘Žđ?‘˜đ?‘” đ?‘?đ?œ€2 + 2đ?œ€3 2

((đ?‘?đ?‘˜đ?‘” ) − 2)

5/2

,

Theorem 13. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a timelike pseudospherical đ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by đ?‘? đ?‘Ž 0 [ ] √2 √2 [ ] [ ] [ ] [ ] −đ?‘?đ?‘˜ −đ?‘Žđ?‘˜ −đ?‘Ž đ?‘” đ?‘” đ?›˝ [ ] đ?›ź ] [đ?‘‡đ?›˝ ] = [ [ √2đ?‘˜đ?‘”2 − đ?‘Ž2 √2đ?‘˜đ?‘”2 − đ?‘Ž2 √2đ?‘˜đ?‘”2 − đ?‘Ž2 ] [đ?‘‡] , [ ] đ?œ‰ ][ ] [ đ?œ‰đ?›˝ ] [ [ ] [ −2đ?‘˜ ] [ ] −đ?‘Žđ?‘? −đ?‘Ž2 đ?‘” [ ] 4đ?‘˜đ?‘”2 − 2đ?‘Ž2 √4đ?‘˜đ?‘”2 − 2đ?‘Ž2 √4đ?‘˜đ?‘”2 − 2đ?‘Ž2 √ [ ] (62) and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝

=

−2đ?‘˜đ?‘” đ?œ€1 − đ?‘?đ?‘Žđ?œ€2 + đ?‘Ž2 đ?œ€3 (2đ?‘˜đ?‘”2 − đ?‘Ž2 )

5/2

,

(63)

where đ?œ€1 = 2đ?‘Žđ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + 2đ?‘?đ?‘˜đ?‘”3 − đ?‘Ž2 đ?‘?đ?‘˜đ?‘” ,

(60)

đ?œ€2 = 2đ?‘Žđ?‘˜đ?‘”4 − (đ?‘Ž3 + 2đ?‘Ž) đ?‘˜đ?‘”2 + đ?‘Ž3 + đ?‘Ž2 đ?‘?đ?‘˜đ?‘”ó¸€ ,

(64)

đ?œ€3 = đ?‘Ž3 đ?‘˜đ?‘”ó¸€ + 2đ?‘?đ?‘˜đ?‘”4 − đ?‘Ž2 đ?‘?đ?‘˜đ?‘”2 ,

where đ?‘Ž, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 − đ?‘?2 = 2, and 2đ?‘˜đ?‘”2 − đ?‘Ž2 > 0 for all đ?‘ .

đ?œ€1 = đ?‘?3 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” − đ?‘Žđ?‘?2 đ?‘˜đ?‘”2 + 2đ?‘Ž, đ?œ€2 = − đ?‘Žđ?‘?2 đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + đ?‘?3 đ?‘˜đ?‘”4 − (đ?‘?3 + 2đ?‘?) đ?‘˜đ?‘”2 + 2đ?‘?, đ?œ€3 = 2đ?‘?đ?‘˜đ?‘”ó¸€ âˆ’ đ?‘Žđ?‘?2 đ?‘˜đ?‘”3 + 2đ?‘Žđ?‘˜đ?‘” , đ?‘Ž, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 + đ?‘?2 = 2, and (đ?‘?đ?‘˜đ?‘” )2 − 2 > 0 for all đ?‘ .

(61)

Theorem 14. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 is a timelike pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by

đ?‘? đ?‘? đ?‘Ž [ ] √ √ √ 3 3 3 [ ] [ ] [ ] đ?‘Ž − đ?‘?đ?‘˜đ?‘” −đ?‘?đ?‘˜đ?‘” [ ] −đ?‘? [ ] đ?›˝ [ ] đ?›ź 2 2 2 ] [ [đ?‘‡đ?›˝ ] = [ √ đ?‘?2 (đ?‘˜2 − 1) − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) √ đ?‘?2 (đ?‘˜đ?‘”2 − 1) − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) √ đ?‘?2 (đ?‘˜đ?‘”2 − 1) − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) ] [đ?‘‡] đ?‘” [ ] đ?œ‰ ][ ] [ đ?œ‰đ?›˝ ] [ [ ] [ ] 2 2 −đ?‘? đ?‘˜đ?‘” − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘? − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘Ž − đ?‘? [ ] đ?‘Žđ?‘?đ?‘˜đ?‘” − đ?‘?đ?‘? [ ] [ ] 2 2 2 √ 3đ?‘?2 (đ?‘˜đ?‘”2 − 1) − 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) √ 3đ?‘?2 (đ?‘˜đ?‘”2 − 1) − 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) √ 3đ?‘?2 (đ?‘˜đ?‘”2 − 1) − 3(đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) [ ]

(65)

8

Journal of Applied Mathematics

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads

fully in pseudosphere đ?‘†12 . The curve đ?›˝ is pseudospherical đ?›źđ?œ‰Smarandache curve of đ?›ź, if it is given by

đ?‘˜đ?‘”đ?›˝ = (((đ?‘Ž2 − 3) đ?‘˜đ?‘” − đ?‘Žđ?‘?) đ?œ€1 + (−đ?‘?đ?‘? + đ?‘Žđ?‘?đ?‘˜đ?‘” ) đ?œ€2

đ?›˝ (đ?‘ ) =

2

+ (đ?‘? + 3 − đ?‘Žđ?‘?đ?‘˜đ?‘” ) đ?œ€3 ) ) , (66) where đ?œ€1 = (3 − đ?‘Ž2 ) đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”ó¸€ + (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”3

+ đ?‘Ž2 đ?‘?đ?‘˜đ?‘”ó¸€ + (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”4

(69)

where đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 + đ?‘?2 = 2.

đ?›˝ (đ?‘ ) =

− 2đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”3 + (đ?‘?3 − 3đ?‘?) đ?‘˜đ?‘”2 − 2đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘” + đ?‘? (đ?‘?2 − 3) ,

1 (đ?‘Žđ?‘‡ (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √2

(70)

where đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 − đ?‘?2 = 2.

đ?œ€3 = − đ?‘Žđ?‘?đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” + (đ?‘?2 − 3) đ?‘?đ?‘˜đ?‘”ó¸€ + (đ?‘Ž2 − 3) đ?‘?đ?‘˜đ?‘”4 + (đ?‘Žđ?‘? − 3đ?‘Žđ?‘?

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?‘‡ (đ?‘ )) , √2

Definition 18. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 and đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere đ?‘†12 . The curve đ?›˝ is pseudospherical đ?‘‡đ?œ‰Smarandache curve of đ?›ź, if it is given by

đ?œ€2 = (3 − đ?‘Ž2 ) đ?‘?đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘”2 + (2đ?‘Žđ?‘?2 − đ?‘Žđ?‘?2 ) đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘”

2

Definition 17. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 and đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere đ?‘†12 . The curve đ?›˝ is pseudospherical đ?›źđ?‘‡Smarandache curve of đ?›ź, if it is given by đ?›˝ (đ?‘ ) =

+ (đ?‘Žđ?‘?2 − 3đ?‘Žđ?‘?2 ) đ?‘˜đ?‘”2 + (đ?‘?2 đ?‘? − 3đ?‘Ž2 đ?‘?) đ?‘˜đ?‘” + đ?‘Ž (đ?‘?2 − 3) ,

2

(68)

where đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 − đ?‘?2 = 2.

−1 2 5/2

Ă— ((đ?‘?2 (đ?‘˜đ?‘”2 − 1) − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) )

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √2

Definition 19. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 and đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike and null curve, respectively, lying fully in pseudosphere đ?‘†12 . The curve đ?›˝ is pseudospherical đ?›źđ?‘‡đ?œ‰Smarandache curve of đ?›ź, if it is given by

) đ?‘˜đ?‘”3

+ (đ?‘?2 đ?‘? − 3đ?‘Ž2 đ?‘?) đ?‘˜đ?‘”2 + (đ?‘?2 − 3) đ?‘Žđ?‘˜đ?‘” , (67) đ?‘Ž, đ?‘?, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 + đ?‘?2 − đ?‘?2 = 3, and đ?‘?2 (đ?‘˜đ?‘”2 − 1) − (đ?‘Ž − đ?‘?đ?‘˜đ?‘” )2 > 0 for all đ?‘ . Corollary 15. If đ?›ź is a spacelike geodesic curve on pseudosphere đ?‘†12 in Minkowski 3-space đ??¸13 , then (1) the spacelike and timelike pseudospherical đ?›źđ?‘‡-Smarandache curves are also geodesic on đ?‘†12 ; (2) the spacelike and timelike pseudospherical đ?‘‡đ?œ‰ and đ?›źđ?‘‡đ?œ‰Smarandache curves have constant geodesic curvatures on đ?‘†12 ;

đ?›˝ (đ?‘ ) =

In this section, we give definitions of null pseudospherical Smarandache curves which are analogous to the definitions of nonnull pseudospherical Smarandache curves of đ?›ź, given in Section 3. Definition 16. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 and đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike and a null curve, respectively, lying

(71)

where đ?‘Ž, đ?‘?, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 + đ?‘?2 − đ?‘?2 = 3. Theorem 20. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . Then the null pseudospherical đ?›źđ?œ‰-Smarandache curve of đ?›ź does not exist. Proof. Assume that there exists null pseudospherical đ?›źđ?œ‰Smarandache curve of đ?›ź. Differentiating (68) with respect to đ?‘ and using (4) we obtain đ?›˝ó¸€ (đ?‘ ) =

(3) the spacelike pseudospherical đ?›źđ?œ‰-Smarandache curve has constant geodesic curvature on đ?‘†12 .

4. Null Pseudospherical Smarandache Curves in Minkowski 3-Space

1 (đ?‘Žđ?›ź (đ?‘ ) + đ?‘?đ?‘‡ (đ?‘ ) + đ?‘?đ?œ‰ (đ?‘ )) , √3

đ?‘Ž − đ?‘?đ?‘˜đ?‘” √2

�.

(72)

This means that a null vector đ?‘‡đ?›˝ = đ?›˝ó¸€ is collinear with a spacelike vector đ?‘‡, which is a contradiction. Theorem 21. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ is a null pseudospherical đ?›źđ?‘‡-Smarandache curve of đ?›ź, then đ?›ź has constant geodesic curvature given by đ?‘˜đ?‘”2 =

2 , 2 − đ?‘Ž2

đ?‘Ž ∈ đ?‘…0 , đ?‘Ž2 < 2.

(73)

Journal of Applied Mathematics

9

Proof. Differentiating (69) with respect to đ?‘ and using (4) we obtain 1 đ?›˝ó¸€ (đ?‘ ) = (−đ?‘?đ?›ź + đ?‘Žđ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , (74) √2 and consequently 1 (−đ?‘?đ?›ź + đ?‘Žđ?‘‡ − đ?‘?đ?‘˜đ?‘” đ?œ‰) , đ?‘‡đ?›˝ = (75) √2 where đ?‘Ž, đ?‘? ∈ đ?‘…0 , and đ?‘Ž2 + đ?‘?2 = 2. The condition â&#x;¨đ?‘‡đ?›˝ , đ?‘‡đ?›˝ â&#x;Š = 0 implies 2 (76) , đ?‘Ž ∈ đ?‘…0 , đ?‘Ž2 < 2, đ?‘˜đ?‘”2 = 2 − đ?‘Ž2 which proves the theorem.

20

đ?‘˜đ?‘” =

2đ?‘Žđ?‘? Âą √4đ?‘Ž2 đ?‘?2 − 4 (đ?‘?2 − đ?‘?2 ) (đ?‘Ž2 + đ?‘?2 ) 2 (đ?‘Ž2 − 3)

,

(78)

0

x

10

20 20

(79)

The orthonormal Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} along the curve đ?›ź is given by đ?›ź (đ?‘ ) = (

đ?‘ 2 đ?‘ 2 , đ?‘ , 1 − ) , 2 2

đ?‘‡ (đ?‘ ) = đ?›źó¸€ (đ?‘ ) = (đ?‘ , 1, −đ?‘ ) , đ?œ‰ (đ?‘ ) = − đ?›ź (đ?‘ ) Ă— đ?‘‡ (đ?‘ ) = (−1 −

(80) đ?‘ 2 đ?‘ 2 , −đ?‘ , ) . 2 2

−20

(81)

Case 1. Taking đ?‘Ž = √3 and đ?‘? = 1 and using (8), we obtain that pseudospherical đ?›źđ?œ‰-Smarandache curve đ?›˝ is given by (see Figure 2)

=

đ?‘ 2 đ?‘ 2 , đ?‘ , 1 − ) . 2 2

−10

đ?‘˜đ?‘” (đ?‘ ) = det (đ?›ź (đ?‘ ) , đ?‘‡ (đ?‘ ) , đ?‘‡ó¸€ (đ?‘ )) = −1.

Corollary 24. There are no spacelike pseudospherical geodesic curves whose pseudospherical đ?›źđ?‘‡, đ?‘‡đ?œ‰, and đ?›źđ?‘‡đ?œ‰-Smarandache curves are the null curves.

đ?›ź (đ?‘ ) = (

0 y

In particular, the geodesic curvature đ?‘˜đ?‘” of the curve đ?›ź has the form

đ?›˝ (đ?‘ )

Example 1. Let � be a unit speed spacelike curve lying on pseudosphere �12 in the Minkowski 3-space E31 with parameter equation (see Figure 1)

10

Figure 1: The curve � on �12 .

where đ?‘Ž, đ?‘?, đ?‘? ∈ đ?‘…0 , đ?‘Ž2 + đ?‘?2 − đ?‘?2 = 3, and đ?‘Ž2 ≠ 3.

5. Examples

−20

−10

Theorem 22. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ is a null pseudospherical đ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then đ?›ź has constant geodesic curvature given by đ?‘Ž2 (77) , đ?‘Ž ∈ đ?‘…0 , đ?‘Ž2 > 2. 2 Theorem 23. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ?‘†12 be a unit speed spacelike curve lying fully in đ?‘†12 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ is a null pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then đ?›ź has constant geodesic curvature given by

z 0 −10

−20

The next two theorems can be proved in a similar way, so we omit their proofs.

đ?‘˜đ?‘”2 =

10

(√3 − 1) 2 √2 (√3 − 1) 2 ( đ?‘ − 1, (√3 − 1) đ?‘ , √3 − đ?‘ ). 2 2 2 (82)

It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š > 0, which means that đ?›˝ is a spacelike curve. According to Theorem 7, its Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 √6 [ 2 0 2 ] ] đ?›ź [ đ?›˝ ] [ [đ?‘‡đ?›˝ ] = [ 0 1 0 ] [đ?‘‡] , ] [ ] [ [ đ?œ‰đ?›˝ ] [ √2 √6 ] [ đ?œ‰ ] 0 2 ] [ 2

(83)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘? − đ?‘Žđ?‘˜đ?‘” đ?‘˜đ?‘”đ?›˝ = 󾄨󾄨 󾄨 = 1. 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 󾄨 󾄨

(84)

10

Journal of Applied Mathematics

z

20

20 10

10

z

0

−10

0

−10

−20

−20

−20

−20

−10

0

−10

x

0

10

20

10

0 y

10

20

−10

20

−20

Figure 2: The spacelike đ?›źđ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

10

0 y

−10

−20

đ?›˝ (đ?‘ ∗ (đ?‘ )) =

−20 −10 0

2

√2 đ?‘ − 2đ?‘ đ?‘ ( , đ?‘ − 1, − + đ?‘ ) . 2 2 2 ó¸€

−

0 ] [ [ ] đ?›ź đ?›˝ [ ] [đ?‘‡đ?›˝ ] = [ ] [đ?‘‡] , 1 [ ] [ ] đ?œ‰ đ?œ‰ [ đ?›˝ ] [ √2 √2 ][ ] − − −√2 2 [ 2 ]

−10

20

10

0

−10

−20

y

Figure 4: The spacelike đ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

(87)

Case 3. Taking đ?‘Ž = 2 and đ?‘? = √2, from (10) we find that the pseudospherical đ?‘‡đ?œ‰-Smarandache curve is given by (see Figure 4) đ?›˝ (đ?‘ ∗ (đ?‘ )) =

0

−20

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ = −√2.

10 x

(86)

20

20

ó¸€

√2 2 1

z 10

(85)

It can be easily checked that â&#x;¨đ?›˝ (đ?‘ ), đ?›˝ (đ?‘ )â&#x;Š > 0, which means that đ?›˝ is a spacelike curve. According to Theorem 8, its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 2 1

20

Figure 3: The spacelike ��-pseudospherical Smarandache curve � and the curve � on �12 .

Case 2. Taking đ?‘Ž = 1 and đ?‘? = −1 and using (9) we get that pseudospherical đ?›źđ?‘‡-Smarandache curve is given by (see Figure 3) 2

x

√2 √2 2 √2 (− đ?‘ 2 + 2đ?‘ − √2, 2 − √2đ?‘ , đ?‘ − 2đ?‘ ) . 2 2 2 (88)

It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š > 0, which means that đ?›˝ is a spacelike curve. According to Theorem 9, its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 đ?›˝ 0 1 đ?›ź [đ?‘‡đ?›˝ ] = [−√2 −1 −√2] [đ?‘‡] , [ đ?œ‰đ?›˝ ] [ −1 −√2 −2 ] [ đ?œ‰ ]

(89)

Journal of Applied Mathematics

z

11 20

20 10

10

0

z

0

−10 −20

−10

−20

−20 −20 −10 −10 0

x 0

10

20

10

0 y

−10

−20

10

20 20

Figure 5: The spacelike đ?›źđ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

(90)

10

−20

−10

0 y

20

Figure 6: Special spacelike pseudospherical Smarandache curves of � and the curve � on �12 .

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ has the following form: đ?‘˜đ?‘”đ?›˝ = 1.

20

−20 z −10 0 10 −20

Case 4. Taking đ?‘Ž = √2, đ?‘? = √2, and đ?‘? = 1 and using (11), we find that the pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve đ?›˝ has parameter equation (see Figure 5): đ?›˝ (đ?‘ ∗ (đ?‘ )) =

−10

(√2 + 1) 2 1 ( đ?‘ + √2đ?‘ + 1, (1 + √2) đ?‘ + √2, √3 2 −

(1 + √2) 2

x

0

x

10

đ?‘ 2 − √2đ?‘ + √2) . (91)

It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š > 0, which means that đ?›˝ is a spacelike curve. By Theorem 10, its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √6 √3 √6 [ ] 3 3 3 [ ] đ?›ź đ?›˝ [ ] [đ?‘‡đ?›˝ ] = [ −√2 − 2 √2 − 2 ] [đ?‘‡] , (92) 1 − [ ] [ ] [ đ?œ‰đ?›˝ ] [ 2√6 + 3√3 √6 3√6 + 2√3 ] [ đ?œ‰ ] − − 3 3 3 [ ] while the corresponding geodesic curvature lowing form: đ?‘˜đ?‘”đ?›˝ = −1200√2 − 1697.

đ?‘˜đ?‘”đ?›˝

has the fol-

(93)

−20

−10

0 y

20

10

20

Figure 7: The spacelike circle � on �12 .

Special spacelike Smarandache curves of đ?›ź and the curve đ?›ź on đ?‘†12 are shown in Figure 6. Example 2. Let us consider a unit speed spacelike circle đ?›ź lying on pseudosphere đ?‘†12 in the Minkowski 3-space E31 with parameter equation (see Figure 7): đ?›ź (đ?‘ ) = (cosh đ?‘ , sinh đ?‘ , √2) .

(94)

12

Journal of Applied Mathematics −20 −20 −10

z

−10

y 0

0

10

20

10 −20

20 −20

−10

x

−10 −20

0

−10 10

0

20

x

0 10

z

20

10 −20

−10

0 y

10

20

20

Figure 8: The timelike ��-pseudospherical Smarandache curve � and the curve � on �12 .

The orthonormal Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} along the curve đ?›ź is given by đ?›ź (đ?‘ ) = (cosh đ?‘ , sinh đ?‘ , √2) , đ?‘‡ (đ?‘ ) = đ?›źó¸€ (đ?‘ ) = (sinh đ?‘ , cosh đ?‘ , 0) ,

(95)

đ?œ‰ (đ?‘ ) = −đ?›ź (đ?‘ ) Ă— đ?‘‡ (đ?‘ ) = (−√2 cosh đ?‘ , −√2 sinh đ?‘ , −1) . In particular, the geodesic curvature đ?‘˜đ?‘” of đ?›ź reads đ?‘˜đ?‘” (đ?‘ ) = det (đ?›ź (đ?‘ ) , đ?‘‡ (đ?‘ ) , đ?‘‡ó¸€ (đ?‘ )) = −√2.

(96)

Case 1. Taking đ?‘Ž = (√2/2) and đ?‘? = (√6/2) and using (9), we obtain that pseudospherical đ?›źđ?‘‡-Smarandache curve đ?›˝ is given by (see Figure 8) √2 √3 √3 1 1 đ?›˝ (đ?‘ ) = ( cosh đ?‘ + sinh đ?‘ , sinh đ?‘ + cosh đ?‘ , ). 2 2 2 2 2 (97) It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š < 0, which means that đ?›˝ is a timelike curve. According to Theorem 12, its Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √3 1 0] [ 2 [ 2 ] [ ] đ?›ź đ?›˝ [ √6 √2 ] [đ?‘‡đ?›˝ ] = [− [ ] √ 3] [ ] đ?‘‡ , 2 2 [ ] đ?œ‰ [ đ?›˝] [ ] [đ?œ‰] [ 3 ] √3 √2 2 [ 2 ]

(98)

Figure 9: The timelike đ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ =

−đ?‘˜đ?‘” đ?‘?2 đ?œ€1 + đ?‘Žđ?‘˜đ?‘” đ?‘?đ?œ€2 + 2đ?œ€3 2

((đ?‘?đ?‘˜đ?‘” ) − 2)

5/2

= −10.

(99)

Case 2. Taking đ?‘Ž = √3 and đ?‘? = 1 and using (10), we obtain that pseudospherical đ?‘‡đ?œ‰-Smarandache curve đ?›˝ is given by (see Figure 9) đ?›˝ (đ?‘ ) = (√3 sinh đ?‘ − √2 cosh đ?‘ , √3 cosh đ?‘ − √2 sinh đ?‘ , −1) . (100) It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š < 0, which means that đ?›˝ is a timelike curve. According to Theorem 13, its Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 √6 [ 0 2 2 ] [ ] đ?›ź đ?›˝ [ ] [đ?‘‡đ?›˝ ] = [−√3 √2 √6 ] [đ?‘‡] , [ ] [ ] [ đ?œ‰đ?›˝ ] [ √6 3√2 ] [ đ?œ‰ ] 2√2 − − 2 2 ] [

(101)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ = 11.

(102)

Case 3. Taking đ?‘Ž = −√2, đ?‘? = √2, and đ?‘? = 1 and using (11), we obtain that pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve đ?›˝ is given by (see Figure 10) đ?›˝ (đ?‘ ) = (√2 sinh đ?‘ − 2√2 cosh đ?‘ , √2 cosh đ?‘ − 2√2 sinh đ?‘ , −3) . (103)

Journal of Applied Mathematics

13 y

y −20

−20

0

−10

10

20

−20

z

0

0

10

20

0 10

10 20

20

−20

−20 −10

−10 x

−10

−10

−10 z

−20

x

0

0 10

10

20

20

Figure 10: The timelike đ?›źđ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š < 0, which means that đ?›˝ is a timelike curve. According to Theorem 14, its Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √6 √6 √3 [− 3 3 3 ] [ ] đ?›ź đ?›˝ [ ] [đ?‘‡đ?›˝ ] = [ −1 0 −√2 ] [đ?‘‡] , [ ] [ ] [ đ?œ‰đ?›˝ ] [ 2√3 √3 √6 ] [ đ?œ‰ ] − 3 3 ] [ 3

Figure 11: Special timelike pseudospherical Smarandache curves of � and the curve � on �12 .

−20 −10 z

0 10

20

−20

(104) −10

0

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ = −5√2.

Example 3. Let us consider a unit speed spacelike circle đ?›ź lying on pseudosphere đ?‘†12 in the Minkowski 3-space E31 with parameter (94). Then its geodesic curvature is given by (96). Case 1. If đ?›˝ is null pseudospherical đ?›źđ?‘‡-Smarandache curve of đ?›ź, then according to Theorem 21 the curve đ?›ź has nonzero constant geodesic curvature given by đ?‘˜đ?‘”2 =

2 , 2 − đ?‘Ž2

đ?‘Ž ∈ đ?‘…0 , đ?‘Ž2 < 2.

10

(105)

Special timelike Smarandache curves of � and the curve � on �12 are shown in Figure 11.

(106)

The last relation together with (96) implies đ?‘Ž = 1. By Definition 17 there holds đ?‘Ž2 +đ?‘?2 = 2 and therefore we can take

x

−20

−10

0 y

10

20

20

Figure 12: The null đ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

đ?‘? = 1. Finally, by using (69) we obtain that pseudospherical đ?›źđ?‘‡-Smarandache curve đ?›˝ of đ?›ź is given by (see Figure 12) đ?›˝ (đ?‘ ) =

√2 (sinh đ?‘ + cosh đ?‘ , cosh đ?‘ + sinh đ?‘ , √2) . 2

(107)

It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š = 0, which means that đ?›˝ is a null straight line.

14

Journal of Applied Mathematics y −20

−20

−10

0

y 10

20

−20

10

0

−10

−20

−10

−10 z

20

z

0

0 10

10

20 20

20 −20

10

−10 x

x

0

0 −10

10

−20

20

Figure 13: The null đ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

Figure 14: The null đ?›źđ?‘‡đ?œ‰-pseudospherical Smarandache curve đ?›˝ and the curve đ?›ź on đ?‘†12 .

Case 2. If đ?›˝ is null pseudospherical đ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then according to Theorem 22 the curve đ?›ź has nonzero constant geodesic curvature given by

Conflict of Interests

đ?‘Ž2 (108) , đ?‘Ž ∈ đ?‘…0 , đ?‘Ž2 > 2. 2 The last relation together with (96) implies đ?‘Ž = 2. By Definition 18 there holds đ?‘Ž2 −đ?‘?2 = 2 and therefore we can take đ?‘? = √2. Finally, by using (70) we obtain that pseudospherical đ?‘‡đ?œ‰-Smarandache curve đ?›˝ of đ?›ź is given by (see Figure 13) đ?‘˜đ?‘”2 =

đ?›˝ (đ?‘ ) = (2 sinh đ?‘ − 2 cosh đ?‘ , 2 cosh đ?‘ − 2 sinh đ?‘ , −√2) . (109) It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š = 0, which means that đ?›˝ is a null straight line. Case 3. If đ?›˝ is null pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve of đ?›ź, then according to Theorem 23 the curve đ?›ź has nonzero constant geodesic curvature given by đ?‘˜đ?‘” =

2đ?‘Žđ?‘? Âą √4đ?‘Ž2 đ?‘?2 − 4 (đ?‘?2 − đ?‘?2 ) (đ?‘Ž2 + đ?‘?2 ) 2 (đ?‘Ž2 − 3)

.

(110)

The last relation together with (96) implies đ?‘Ž = 2, đ?‘? = √3. By Definition 19 there holds đ?‘Ž2 + đ?‘?2 − đ?‘?2 = 3 and therefore we can take đ?‘? = −2. Finally, by using (71) we obtain that pseudospherical đ?›źđ?‘‡đ?œ‰-Smarandache curve đ?›˝ of đ?›ź is given by (see Figure 14) đ?›˝ (đ?‘ ) = ((2 + 2√2) cosh đ?‘ + √3 sinh đ?‘ , (2 + 2√2) sinh đ?‘ +√3 cosh đ?‘ , 2√2 + 2) . (111) It can be easily checked that â&#x;¨đ?›˝ó¸€ (đ?‘ ), đ?›˝ó¸€ (đ?‘ )â&#x;Š = 0, which means that đ?›˝ is a null straight line.

The authors declare that there is no conflict of interests regarding the publication of this paper.

References [1] C. Ashbacher, “Smarandache geometries,â€? Smarandache Notions Journal, vol. 8, no. 1–3, pp. 212–215, 1997. [2] M. Turgut and S. YÄąlmaz, “Smarandache curves in Minkowski space-time,â€? International Journal of Mathematical Combinatorics, vol. 3, pp. 51–55, 2008. [3] A. T. Ali, “Special smarandache curves in the euclidean space,â€? International Journal of Mathematical Combinatorics, vol. 2, pp. 30–36, 2010. [4] E. Betul Koc Ozturk, U. Ozturk, I. Ilarslan, and E. Neˇsovi´c, “On pseudohyperbolical Smarandache curves in Minkowski 3space,â€? International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 658670, 7 pages, 2013. [5] T. K¨orpinar and E. Turhan, “A new approach on timelike biharmonic slant helices according to Bishop frame in Lorentzian Heisenberg group đ??ťđ?‘’đ?‘–đ?‘ 3 ,â€? Kochi Journal of Mathematics, vol. 6, pp. 139–148, 2011. [6] T. K¨orpinar and E. Turhan, “Characterization of Smarandache M1 M2 curves of spacelike biharmonic đ??ľ-slant helices according to Bishop frame in E(1, 1),â€? Advanced Modeling and Optimization, vol. 14, no. 2, pp. 327–333, 2012. [7] K. Tas¸k¨opr¨u and M. Tosun, “Smarandache curves on đ?‘†2 ,â€? Boletim da Sociedade Paranaense de Matem´atica, vol. 32, no. 1, pp. 51–59, 2014. ˇ curovi´c, “Some characteriza[8] M. Petrovi´c-Torgaˇsev and E. Su´ tions of the Lorentzian spherical timelike and null curves,â€? Matematichki Vesnik, vol. 53, no. 1-2, pp. 21–27, 2001. [9] B. O’Neill, Semi-Riemannian Geometry, vol. 103 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1983.

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