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## Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$

#### Fatma Ateş [1] , İsmail Gok [2] , Faik nejat Ekmekci [3] , Yusuf Yaylı [4]

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The aim of this paper is to introduce inclined curves according to parallel transport frame. This paper begins by defined a vector field $D$ called Darboux vector field of an inclined curve in $E^{4}$. It will then go on to an alternative characterization for the inclined curves

$\alpha :I\subset \mathbb{R} \longrightarrow E^{4}\text{ is an inclined curve}\Leftrightarrow k_{1}(s)\int k_{1}(s)ds+k_{2}(s)\int k_{2}(s)ds+k_{3}(s)\int k_{3}(s)ds=0$

where $k_{1}(s)$, $k_{2}(s),$ $k_{3}(s)$ are the principal curvature functions according to parallel transport frame of the curve $\alpha$ and also, similar characterization for the generalized helices according to Bishop frame in $E^{3}$ is given by

$\alpha :I\subset \mathbb{R} \longrightarrow E^{3}\text{ is a generalized helix}\Leftrightarrow k_{1}(s)\int k_{1}(s)ds+k_{2}(s)\int k_{2}(s)ds=0$

where $k_{1}(s)$, $k_{2}(s)$ are the principal curvature functions according to Bishop frame of the curve $\alpha$. These curves have illustrated some examples and draw their figures with use of Mathematica programming language. Also, it is given an example for the inclined curve in $E^{4}$ and showed that the above condition is satisfied for this curve.

Parallel transport frame, Inclined curve, Bishop frame, Generalized helix
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Primary Language tr Engineering Articles Orcid: 0000-0002-3529-1077Author: Fatma Ateş (Primary Author)Country: Turkey Author: İsmail Gok Author: Faik nejat Ekmekci Author: Yusuf Yaylı Publication Date: April 15, 2019
 Bibtex @research article { konuralpjournalmath525932, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2019}, volume = {7}, pages = {16 - 24}, doi = {}, title = {Characterizations of Inclined Curves According to Parallel Transport Frame in \$E\^\{4\}\$ and Bishop Frame in \$E\^\{3\}\$}, key = {cite}, author = {Ateş, Fatma and Gok, İsmail and Ekmekci, Faik nejat and Yaylı, Yusuf} } APA Ateş, F , Gok, İ , Ekmekci, F , Yaylı, Y . (2019). Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$. Konuralp Journal of Mathematics (KJM), 7 (1), 16-24. Retrieved from http://dergipark.org.tr/konuralpjournalmath/issue/31492/525932 MLA Ateş, F , Gok, İ , Ekmekci, F , Yaylı, Y . "Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$". Konuralp Journal of Mathematics (KJM) 7 (2019): 16-24 Chicago Ateş, F , Gok, İ , Ekmekci, F , Yaylı, Y . "Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$". Konuralp Journal of Mathematics (KJM) 7 (2019): 16-24 RIS TY - JOUR T1 - Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$ AU - Fatma Ateş , İsmail Gok , Faik nejat Ekmekci , Yusuf Yaylı Y1 - 2019 PY - 2019 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 16 EP - 24 VL - 7 IS - 1 SN - -2147-625X M3 - UR - Y2 - 2019 ER - EndNote %0 Konuralp Journal of Mathematics (KJM) Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$ %A Fatma Ateş , İsmail Gok , Faik nejat Ekmekci , Yusuf Yaylı %T Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$ %D 2019 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 7 %N 1 %R %U ISNAD Ateş, Fatma , Gok, İsmail , Ekmekci, Faik nejat , Yaylı, Yusuf . "Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$". Konuralp Journal of Mathematics (KJM) 7 / 1 (April 2019): 16-24. AMA Ateş F , Gok İ , Ekmekci F , Yaylı Y . Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$. Konuralp J. Math.. 2019; 7(1): 16-24. Vancouver Ateş F , Gok İ , Ekmekci F , Yaylı Y . Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$. Konuralp Journal of Mathematics (KJM). 2019; 7(1): 24-16.