On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

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Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 658670, 7 pages http://dx.doi.org/10.1155/2013/658670

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

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

Correspondence should be addressed to Ufuk Ozturk; uuzturk@asu.edu Received 10 March 2013; Accepted 7 May 2013 Academic Editor: B. N. Mandal Copyright Š 2013 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. We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves.

1. Introduction In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the problem of characterization of a regular curve. In the solution of the problem, the curvature functions đ?œ… and đ?œ? of a regular curve have an effective role. It is known that we can determine the shape and size of a regular curve by using its curvatures đ?œ… and đ?œ?. Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that 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. If the position vector of a regular curve đ?›ź is composed by the Frenet frame vectors of another regular curve đ?›˝, then the curve đ?›ź is called a Smarandache curve [2]. Special Smarandache curves in Euclidean and Minkowski spaces are studied by some authors [3–8]. The curves lying on

a pseudohyperbolic space đ??ť02 in Minkowski 3-space E31 are characterized in [9]. In this paper, we define pseudohyperbolical Smarandache curves according to the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that special đ?‘‡đ?œ‰-pseudohyperbolical Smarandache curves do not exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space.

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 Cartesian coordinate system of R31 . Since đ?‘” is an indefinite metric, recall that a nonzero vector đ?‘Ľâƒ— ∈ R31 can have one of the 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 ⃗ and (length) of a vector đ?‘Ľâƒ— ∈ R31 is given by ‖đ?‘Ľâ€–⃗ = √|â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Š|

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International Journal of Mathematics and Mathematical Sciences

two vectors đ?‘Ľâƒ— 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 every đ?‘ ∈ đ??ź [10]. If ‖đ?›źó¸€ (đ?‘ )‖ ≠ 0 for every đ?‘ ∈ đ??ź, then đ?›ź is a regular curve in R31 . A spacelike (timelike) regular curve đ?›ź is parameterized by pseudo-arclength parameter đ?‘ which is given by đ?›ź : đ??ź ⊂ R → R31 , and then the tangent vector đ?›źó¸€ (đ?‘ ) along đ?›ź has unit length, that is, â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = 1(â&#x;¨đ?›źó¸€ (đ?‘ ), đ?›źó¸€ (đ?‘ )â&#x;Š = −1) for all đ?‘ ∈ đ??ź, respectively. For any đ?‘Ľâƒ— = (đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 ) and đ?‘Śâƒ— = (đ?‘Ś1 , đ?‘Ś2 , đ?‘Ś3 ) in the space R31 , the pseudovector product of đ?‘Ľâƒ— and đ?‘Śâƒ— is defined by đ?‘Ľâƒ— Ă— đ?‘Śâƒ— = (−đ?‘Ľ2 đ?‘Ś3 + đ?‘Ľ3 đ?‘Ś2 , đ?‘Ľ3 đ?‘Ś1 − đ?‘Ľ1 đ?‘Ś3 , đ?‘Ľ1 đ?‘Ś2 − đ?‘Ľ2 đ?‘Ś1 ) .

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Remark 1. Let đ?‘Ľâƒ— = (đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ3 ), đ?‘Śâƒ— = (đ?‘Ś1 , đ?‘Ś2 , đ?‘Ś3 ), and đ?‘§âƒ— = (đ?‘§1 , đ?‘§2 , đ?‘§3 ) be vectors in R31 . Then, 󾄨󾄨 đ?‘Ľ1 đ?‘Ľ2 đ?‘Ľ3 󾄨󾄨 (i) â&#x;¨đ?‘Ľâƒ— Ă— đ?‘Ś,⃗ đ?‘§â&#x;Šâƒ— = 󾄨󾄨󾄨 đ?‘Śđ?‘§1 đ?‘Śđ?‘§2 đ?‘Śđ?‘§3 󾄨󾄨󾄨, 󾄨 1 2 3󾄨 (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?‘§,⃗

Definition 3. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 . Then đ?›źđ?œ‰-pseudohyperbolical Smarandache curve đ?›˝ : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 of đ?›ź is defined by đ?›˝ (đ?‘ ⋆ (đ?‘ )) =

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

(6)

Definition 4. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 .Then đ?›źđ?‘‡-pseudohyperbolical Smarandache curve đ?›˝ : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 of đ?›ź is defined by

⃗ đ?‘Ś,⃗ đ?‘Śâ&#x;Šâƒ— + â&#x;¨đ?‘Ľ,⃗ đ?‘Śâ&#x;Šâƒ— , (iii) â&#x;¨đ?‘Ľâƒ— Ă— đ?‘Ś,⃗ đ?‘Ľâƒ— Ă— đ?‘Śâ&#x;Šâƒ— = −â&#x;¨đ?‘Ľ,⃗ đ?‘Ľâ&#x;Šâ&#x;¨ where Ă— is the pseudovector product in the space R31 . Lemma 2. In the Minkowski 3-space R31 , the following properties are satisfied [10]:

(ii) two null vectors are orthogonal if and only if they are linearly dependent; (iii) timelike vector is never orthogonal to a null vector. Pseudohyperbolic space in the Minkowski 3-space R31 is a quadric defined by đ??ť02 = {đ?‘Ľâƒ— ∈ R31 | −đ?‘Ľ12 + đ?‘Ľ22 + đ?‘Ľ32 = −1} .

(3)

Let đ?›ź : đ??ź ⊂ R → đ??ť02 be a regular unit speed curve lying fully in đ??ť02 in R31 . Then its position vector đ?›ź is timelike vector, which implies that tangent vector đ?‘‡âƒ— = đ?›źó¸€ is the unit spacelike vector for all đ?‘ ∈ đ??ź. 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 ó¸€

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

(4)

where đ?‘˜đ?‘” (đ?‘ ) = det(đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?‘‡ó¸€ (đ?‘ )) is the geodesic curvature of đ?›ź on đ??ť02 in R31 and đ?‘ is arc length parameter of đ?‘Žđ?‘™đ?‘?â„Žđ?‘Ž. In particular, the following relations hold: đ?œ‰ Ă— đ?›ź = đ?‘‡.

đ?›˝ (đ?‘ ⋆ (đ?‘ )) =

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

(7)

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

(i) two timelike vectors are never orthogonal;

đ?‘‡ Ă— đ?œ‰ = −đ?›ź,

In this section, we define and investigate pseudohyperbolical Smarandache curves in Minkowski 3-space according to the Sabban frame. Let đ?›ź = đ?›ź(đ?‘ ) and đ?›˝ = đ?›˝(đ?‘ ∗ ) be two regular unit speed curves lying fully in pseudohyperbolic space đ??ť02 in R31 and let {đ?›ź, đ?‘‡, đ?œ‰} and {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } be the moving Sabban frames of these curves, respectively. Then we have the following definitions of pseudohyperbolical Smarandache curves.

where đ?‘Ž, đ?‘? ∈ R0 and đ?‘Ž2 − đ?‘?2 = 2. 2

đ?›ź Ă— đ?‘‡ = đ?œ‰,

3. Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

(5)

Definition 5. Let đ?›ź : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 . Then đ?›źđ?‘‡đ?œ‰ -pseudohyperbolical Smarandache curve đ?›˝ : đ??ź ⊂ đ?‘… ół¨ƒâ†’ đ??ť02 of đ?›ź is defined by đ?›˝ (đ?‘ ⋆ (đ?‘ )) =

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

(8)

where đ?‘Ž, đ?‘?, đ?‘? ∈ R0 and đ?‘Ž2 − đ?‘?2 − đ?‘?2 = 3. Theorem 6. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 . Then đ?‘‡đ?œ‰-pseudohyperbolical Smarandache curve đ?›˝ : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 of đ?›ź does not exist. Proof. Assume that there exists đ?‘‡đ?œ‰-pseudohyperbolical Smarandache curve of đ?›ź. Then it can be written as đ?›˝ (đ?‘ ⋆ (đ?‘ )) =

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

(9)

where đ?‘Ž2 + đ?‘?2 = −2, which is a contradiction. In the theorems which follow, we obtain Sabban frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } and geodesic curvature đ?œ…đ?‘”đ?›˝ of pseudohyperbolical Smarandache curve đ?›˝. Theorem 7. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and

International Journal of Mathematics and Mathematical Sciences the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ R → ół¨ƒ đ??ť02 is đ?›źđ?œ‰pseudohyperbolical Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by 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

(10)

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

đ?‘Ž, đ?‘? ∈ R0 ,

(11)

where đ?‘Ž2 − đ?‘?2 = 2 and đ?œ– = Âą1. Proof. Differentiating (6) with respect to đ?‘ and using (4), we obtain đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡, đ?›˝ó¸€ (đ?‘ ) = ∗ = √2 đ?‘‘đ?‘ đ?‘‘đ?‘ (12) đ?‘‘đ?‘ ∗ đ?‘Ž − đ?‘?đ?‘˜đ?‘” đ?‘‡đ?›˝ đ?‘‡, = √2 đ?‘‘đ?‘ where 󾄨󾄨 󾄨󾄨 đ?‘‘đ?‘ ∗ 󾄨󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 (13) . = √2 đ?‘‘đ?‘ Therefore, the unit spacelike tangent vector of the curve đ?›˝ is given by đ?‘‡đ?›˝ = đ?œ–đ?‘‡,

(14)

3 Theorem 8. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 is đ?›źđ?‘‡pseudohyperbolical 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?‘˜đ?‘” ) √ 2 + (đ?‘?đ?‘˜đ?‘” ) ] = [ 2 + (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?‘” ) √ 2 + (đ?‘?đ?‘˜đ?‘” ) [ ] đ?›ź Ă— [đ?‘‡] , [đ?œ‰] and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝

đ?‘‘đ?‘ ∗

= đ?œ– (đ?›ź + đ?‘˜đ?‘” đ?œ‰) ,

đ?‘‘đ?‘ and from (13) and (15) we get

đ?‘Ž đ?‘? đ?›ź+đ?œ– đ?œ‰ √2 √2

5/2

,

đ?‘Ž, đ?‘? ∈ R0 ,

(20)

2

2

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

(21)

2

đ?œ€3 = −đ?‘?3 đ?‘˜đ?‘”2 đ?‘˜đ?‘”ó¸€ + (đ?‘Žđ?‘˜đ?‘” + đ?‘?đ?‘˜đ?‘”ó¸€ ) (2 + (đ?‘?đ?‘˜đ?‘” ) ) , and đ?‘Ž2 − đ?‘?2 = 2. Proof. Differentiating (7) with respect to đ?‘ and using (4), we obtain đ?›˝ó¸€ (đ?‘ ) =

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

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

(17)

(22)

where

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

(2 + đ?‘?2 đ?‘˜đ?‘”2 )

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

(15)

√2đ?œ– đ?‘‡đ?›˝ó¸€ = 󾄨󾄨 (16) 󾄨 (đ?›ź + đ?‘˜đ?‘” đ?œ‰) . 󾄨󾄨đ?‘Ž − đ?‘?đ?‘˜đ?‘” 󾄨󾄨󾄨 󾄨 󾄨 On the other hand, from (6) and (14) it can be easily seen that đ?œ‰đ?›˝ = đ?›˝ Ă— đ?‘‡đ?›˝ =đ?œ–

=

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

where

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?‘ . Differentiating (14) with respect to đ?‘ , we find đ?‘‘đ?‘‡đ?›˝ đ?‘‘đ?‘ ∗

(19)

(18)

2

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

(23)

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‰) .

(24)

4

International Journal of Mathematics and Mathematical Sciences Differentiating (24) with respect to đ?‘ , it follows that

đ?‘‘đ?‘‡đ?›˝ đ?‘‘đ?‘ ∗ đ?‘‘đ?‘ ∗ đ?‘‘đ?‘

đ?œ‰đ?›˝ = đ?›˝ Ă— đ?‘‡đ?›˝

1

=

On the other hand, from (7) and (24), it can be easily seen that

2 3/2

(2 + (đ?‘?đ?‘˜đ?‘” ) )

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

(25)

=−

đ?‘?2 đ?‘˜đ?‘” √ 4 + 2(đ?‘?đ?‘˜đ?‘” )

2

đ?›źâˆ’

đ?‘Žđ?‘?đ?‘˜đ?‘” √ 4 + 2(đ?‘?đ?‘˜đ?‘” )

2

√2

�+

√2 2 2

(2 + (đ?‘?đ?‘˜đ?‘” ) )

đ?œ‰.

(28)

From (4) and (23), we get

đ?‘‡đ?›˝ó¸€ =

2

√ 2 + (đ?‘?đ?‘˜đ?‘” )

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

(26)

Hence đ?œ‰đ?›˝ is a unit spacelike vector. Therefore, the geodesic curvature đ?‘˜đ?‘”đ?›˝ of the curve đ?›˝ = ∗ đ?›˝(đ?‘ ) is given by đ?‘˜đ?‘”đ?›˝ = det (đ?›˝, đ?‘‡đ?›˝ , đ?‘‡đ?›˝ó¸€ ) =

where

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

5/2

.

(29)

2

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

đ?œ€2 = −đ?‘Žđ?‘?

đ?‘˜đ?‘” đ?‘˜đ?‘”ó¸€

+ (đ?‘? −

đ?‘?đ?‘˜đ?‘”2 ) (2

2

+ (đ?‘?đ?‘˜đ?‘” ) ) ,

(27)

2

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

đ?‘Ž

[ √3 [ [ [ [ đ?‘? [ đ?›˝ [ 󾄨 󾄨 2 [đ?‘‡đ?›˝ ] = [ [ √ 󾄨󾄨󾄨(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘”2 + 3󾄨󾄨󾄨 󾄨󾄨 󾄨 [ 󾄨 [ đ?œ‰đ?›˝ ] [ [ [ − (đ?‘Ž2 − 3) đ?‘˜đ?‘” + đ?‘Žđ?‘? [ [ [ 󾄨󾄨 󾄨󾄨 2

Theorem 9. Let đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 be a regular unit speed curve lying fully in đ??ť02 with the Sabban frame {đ?›ź, đ?‘‡, đ?œ‰} and the geodesic curvature đ?‘˜đ?‘” . If đ?›˝ : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 is đ?›źđ?‘‡đ?œ‰pseudohyperbolical Smarandache curve of đ?›ź, then its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by

đ?‘? √3

đ?‘? √3

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

] ] ] ] đ?‘?đ?‘˜đ?‘” ] ] đ?›ź 󾄨󾄨 ] 󾄨󾄨 2 ] [đ?‘‡ ] 2 √ 󾄨󾄨󾄨(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘” + 3󾄨󾄨󾄨 ] 󾄨] 󾄨 ] [đ?œ‰] ] 2 ] 3 + đ?‘? − đ?‘Žđ?‘?đ?‘˜đ?‘” ] ] 󾄨󾄨 ] 󾄨󾄨 2

(30)

󾄨 󾄨 2 √ 󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨 √ 󾄨󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨󾄨 √ 󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨 󾄨 󾄨 󾄨] 󾄨 󾄨 [ 󾄨

đ?œ€2 = − ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) đ?‘Žđ?‘˜đ?‘”ó¸€ âˆ’ 3đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” ) (đ?‘Ž − đ?‘?đ?‘˜đ?‘” )

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

2

+ ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘”2 + 3) (đ?‘? − đ?‘?đ?‘˜đ?‘”ó¸€ âˆ’ đ?‘?đ?‘˜đ?‘”2 ) ,

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

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

5/2 −1

) ,

đ?‘Ž, đ?‘?, đ?‘? ∈ R0 , where đ?œ€1 = − ((đ?‘Žđ?‘˜đ?‘” −

đ?‘?) đ?‘Žđ?‘˜đ?‘”ó¸€ 2

−

3đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” ) đ?‘?

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

đ?œ€3 = − ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) đ?‘Žđ?‘˜đ?‘”ó¸€ âˆ’ 3đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” ) đ?‘?đ?‘˜đ?‘” (31)

2

+ ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘”2 + 3) ((đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘˜đ?‘” + đ?‘?đ?‘˜đ?‘”ó¸€ ) (32) and đ?‘Ž2 − đ?‘?2 − đ?‘?2 = 3. Proof. Differentiating (8) with respect to đ?‘ and by using (4), we find đ?‘‘đ?›˝ đ?‘‘đ?‘ ∗ 1 đ?›˝ó¸€ (đ?‘ ) = ∗ (đ?‘?đ?›ź + (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘‡ + đ?‘?đ?‘˜đ?‘” đ?œ‰) , (33) = √3 đ?‘‘đ?‘ đ?‘‘đ?‘

International Journal of Mathematics and Mathematical Sciences

5

and thus ��

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

(34)

where 󾄨󾄨 󾄨󾄨 2 2 󾄨 󾄨 đ?‘‘đ?‘ ∗ √ 󾄨󾄨󾄨(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘” + 3󾄨󾄨󾄨 (35) = . đ?‘‘đ?‘ 3 Âą The geodesic curvature đ?‘˜đ?‘” ≠ (đ?‘Žđ?‘? √đ?‘Ž2 đ?‘?2 − (đ?‘Ž2 − 3)(đ?‘?2 + 3))/(đ?‘Ž2 − 3) for all đ?‘ , since đ?›˝ = đ?›˝(đ?‘ ∗ ) is a unit speed regular curve in R31 . Therefore, the unit spacelike tangent vector of the curve đ?›˝ is given by 1 (đ?‘?đ?›ź + (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘‡ + đ?‘?đ?‘˜đ?‘” đ?œ‰) . đ?‘‡đ?›˝ = 󾄨󾄨 󾄨 2 √ 󾄨󾄨󾄨(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘”2 + 3󾄨󾄨󾄨󾄨 󾄨 󾄨 (36) Differentiating (36) with respect to đ?‘ and from (4) and (35), it follows that đ?‘‡đ?›˝ó¸€ =

√3 2

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

2

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

đ?‘˜đ?‘”đ?›˝ = det (đ?›˝, đ?‘‡đ?›˝ , đ?‘‡đ?›˝ó¸€ ) = (− (đ?‘Žđ?‘? − (đ?‘Ž2 − 3) đ?‘˜đ?‘” ) đ?œ€1 + (đ?‘?đ?‘? − đ?‘Žđ?‘?đ?‘˜đ?‘” ) đ?œ€2

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

3đ?‘˜đ?‘”2

+ ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) −

3đ?‘˜đ?‘”2

+ 3) (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) ,

+ 3) (đ?‘? −

đ?‘?đ?‘˜đ?‘”ó¸€

−

(40)

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

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

đ?œ€2 = − ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) đ?‘Žđ?‘˜đ?‘”ó¸€ âˆ’ 3đ?‘˜đ?‘”ó¸€ đ?‘˜đ?‘” ) (đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) 2

Therefore, the geodesic curvature đ?‘˜đ?‘”đ?›˝ of the curve đ?›˝ = đ?›˝(đ?‘ ) is given by ∗

(37)

where

2

Figure 1: The curve đ?›ź on đ??ť02 (1).

đ?‘?đ?‘˜đ?‘”2 ) ,

5/2 −1

) .

(38) Corollary 10. If đ?›ź : đ??ź ⊂ R ół¨ƒâ†’ đ??ť02 is a geodesic curve on đ??ť02 in Minkowski 3-space E31 , then

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

+ ((đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 3đ?‘˜đ?‘”2 + 3) ((đ?‘Ž − đ?‘?đ?‘˜đ?‘” ) đ?‘˜đ?‘” + đ?‘?đ?‘˜đ?‘”ó¸€ ) . On the other hand, from (8) and (36) it can be easily seen that đ?œ‰đ?›˝ = đ?›˝ Ă— đ?‘‡đ?›˝ =

− (đ?‘Ž2 − 3) đ?‘˜đ?‘” + đ?‘Žđ?‘? 󾄨 󾄨 2 √ 󾄨󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨󾄨 󾄨 󾄨 +

+

(2) đ?›źđ?œ‰-pseudohyperbolic and đ?›źđ?‘‡đ?œ‰-pseudohyperbolic Smarandache curves have constant geodesic curvatures on đ??ť02 .

�

đ?‘?đ?‘? − đ?‘Žđ?‘?đ?‘˜đ?‘” 󾄨 󾄨 2 √ 󾄨󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨󾄨 󾄨 󾄨 3 + đ?‘?2 − đ?‘Žđ?‘?đ?‘˜đ?‘” 󾄨 󾄨 2 √ 󾄨󾄨󾄨󾄨3(đ?‘Žđ?‘˜đ?‘” − đ?‘?) − 9đ?‘˜đ?‘”2 + 9󾄨󾄨󾄨󾄨 󾄨 󾄨

�

4. Examples

đ?œ‰.

Example 1. Let đ?›ź be a unit speed curve lying in pseudohyperbolic space đ??ť02 (1) in the Minkowski 3-space E31 with parameter equation (see Figure 1)

(39) Hence đ?œ‰đ?›˝ is a unit spacelike vector.

(1) đ?›źđ?‘‡-pseudohyperbolic Smarandache curve is also geodesic on đ??ť02 ;

đ?›ź (đ?‘ ) = (

đ?‘ 2 đ?‘ 2 + 1, , đ?‘ ) . 2 2

(41)

6

International Journal of Mathematics and Mathematical Sciences

Figure 2: The đ?›źđ?œ‰-pseudohyperbolical Smarandache curve đ?›˝ and the curve đ?›ź on đ??ť02 (1).

Figure 4: The đ?›źđ?‘‡đ?œ‰-pseudohyperbolical Smarandache curve đ?›˝ and the curve đ?›ź on đ??ť02 (1).

From Theorem 7, its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 0 1 đ?›˝ đ?›ź [đ?‘‡đ?›˝ ] = [ 0 1 0 ] [đ?‘‡] , [ đ?œ‰đ?›˝ ] [ −1 0 √2] [ đ?œ‰ ]

(45)

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

(46)

Case 2. If we take đ?‘Ž = √3, đ?‘? = −1, then from (7) the đ?›źđ?‘‡-pseudohyperbolical Smarandache curve is given by (see Figure 3) Figure 3: The đ?›źđ?‘‡-pseudohyperbolical Smarandache curve đ?›˝ and the curve đ?›ź on đ??ť02 (1).

The orthonormal Sabban frame {đ?›ź(đ?‘ ), đ?‘‡(đ?‘ ), đ?œ‰(đ?‘ )} along the curve đ?›ź is given by đ?‘ 2 đ?‘ 2 đ?›ź (đ?‘ ) = ( + 1, , đ?‘ ) , 2 2 ó¸€

đ?‘‡ (đ?‘ ) = đ?›ź (đ?‘ ) = (đ?‘ , đ?‘ , 1) , đ?œ‰ (đ?‘ ) = đ?›ź (đ?‘ ) Ă— đ?‘‡ (đ?‘ ) = (

(42)

đ?‘ 2 đ?‘ 2 , − 1, đ?‘ ) . 2 2

đ?›˝ (đ?‘ ∗ (đ?‘ )) =

√2 (√3đ?‘ 2 − 2đ?‘ + 2√3, √3đ?‘ 2 − 2đ?‘ , 2√3đ?‘ − 2) . 4 (47)

According to Theorem 8, its frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by √2 √6 − 0 ] [ 2 2 đ?›˝ đ?›ź [ √ √3 ] ] [đ?‘‡] 3 [đ?‘‡đ?›˝ ] = [ , ] [− 1 ] đ?œ‰ [ 3 3 đ?œ‰ ] [ [ đ?›˝] √2 √6 [ ] √6 − [ 6 2 3 ]

(48)

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

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

đ?‘˜đ?‘”đ?›˝ = −1.

đ?‘˜đ?‘” (đ?‘ ) = −1.

Case 3. If đ?‘Ž = 3, đ?‘? = √3, and đ?‘? = √3, then from (7) the đ?›źđ?‘‡đ?œ‰pseudohyperbolical Smarandache curve đ?›˝ is given by (see Figure 4)

(43)

Case 1. If we take đ?‘Ž = 2, đ?‘? = √2, then from (6) the đ?›źđ?œ‰pseudohyperbolical Smarandache curve đ?›˝ is given by (see Figure 2) đ?›˝ (đ?‘ ∗ (đ?‘ )) =

1 √ (( 2 + 1) đ?‘ 2 + 2√2, 2 2

(√2 + 1) đ?‘ − 2, (2√2 + 2) đ?‘ ) .

(44)

đ?›˝ (đ?‘ ∗ (đ?‘ )) =

(49)

1 √ (( 3 + 1) đ?‘ 2 + 2đ?‘ + 2√3, 2 (√3 + 1) đ?‘ 2 +2đ?‘ − 2, (2√3 + 2) đ?‘ + 2) . (50)

International Journal of Mathematics and Mathematical Sciences

Figure 5: Pseudohyperbolical Smarandache curves of đ?›ź and the curve đ?›ź on đ??ť02 (1).

By using Theorem 9, it follows that the frame {đ?›˝, đ?‘‡đ?›˝ , đ?œ‰đ?›˝ } is given by (see Figure 4) √3 1 1 ] [ ] [ ] [√ đ?›˝ [ 3−1 1 − √3 ] đ?›ź [đ?‘‡đ?›˝ ] = [ ] [đ?‘‡] , 1 [ 2 2 ] ] đ?œ‰ [ đ?œ‰ [ đ?›˝] [ ][ ] [ √3 + 1 √3 + 1 ] 1 [ 2 2 ]

(51)

and the corresponding geodesic curvature đ?‘˜đ?‘”đ?›˝ reads đ?‘˜đ?‘”đ?›˝ = −6 (2 + √3) (3 + √3)

1/2

.

(52)

Also, Pseudohyperbolical Smarandache curves of đ?›ź and the curveđ?›ź on đ??ť02 (1) with Figure 5 are shown.

Acknowledgment The authors would like to thank the referees for their helpful suggestions and comments which significantly improved the first version of the 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] K. TaşkoĚˆpruĚˆ and M. Tosun, “Smarandache curves on đ?‘†2 ,â€? Boletim da Sociedade Paranaense de MatemaĚ tica, vol. 32, no. 1, pp. 51–59, 2014. [5] N. Bayrak, O. Bektas, and S. Yuce, “Special Smarandache a Curves in E31 ,â€? In press, http://arxiv.org/abs/1204.5656.

7

[6] T. KoĚˆrpinar 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. 8–15, 2011. [7] T. KoĚˆrpinar and E. Turhan, “Characterization of Smarandache đ?‘€1 đ?‘€2 -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. [8] Y. Yayli and E. Ziplar, “Frenet-serret motion and ruled surfaces with constant slope,â€? International Journal of Physical Sciences, vol. 6, no. 29, pp. 6727–6734, 2011. [9] M. PetrovicĚ -TorgasĚŒev and E. SĚŒucĚ urovicĚ , “Some characterizations of the spacelike, the timelike and the null curves on the pseudohyperbolic space H20 in E31 ,â€? Kragujevac Journal of Mathematics, vol. 22, pp. 71–82, 2000. [10] B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, NY, USA, 1983.

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