Acknowledgement
Supported by : Inonu University
This study has been supported by BAP (Scientific Research Projects) unit under the Project number FDK-2018-1349 at Inonu University, Malatya, TURKEY.
References
- HS. Abdel-Aziz and M.K. Saad, Smarandache Curves Of Some Special Curves in the Galilean 3-Space, Honam Mathematical Journal, 37(2), (2015), 253-264. https://doi.org/10.5831/HMJ.2015.37.2.253
- A.T. Ali, Special Smarandache Curves in the Euclidean Space, Int. J. Math. Comb., 2, (2010), 30-36.
- D. Bakry and M. Emery, Diffusions hypercontractives, Seminaire de probabilites de Strasbourg, Volume 19, (1985), 177-206.
- H. Brauner and W. Kickinger, Baugeometrie 1, Bauverlag, (1977).
- I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7(1), (2006), 1-15.
- M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Func. Anal., 13, (2003), 178-215. https://doi.org/10.1007/s000390300004
-
D.T. Hieu and N.M. Hoang, Ruled minimal surfaces in
$R^3$ with density$e^z$ , Pac. J. Math., 243(2), (2009), 277-285. https://doi.org/10.2140/pjm.2009.243.277 - D.T. Hieu and T.L. Nam, The classification of constant weighted curvature curves in the plane with a log-linear density, Commun. Pure Appl. Anal., 13(4), (2014), 1641-1652. https://doi.org/10.3934/cpaa.2014.13.1641
- J. Hoschek, Liniengeometrie, Bibliograph. Institut, Zurich, (1971).
- A. Kazan and H.B. Karadag, Weighted Minimal And Weighted Flat Surfaces Of Revolution In Galilean 3-Space With Density, Int. J. Anal. Appl., 16(3), (2018), 414-426.
- F. Morgan, Manifolds with Density, Not. Amer. Math. Soc., 52(8), (2005), 853-858.
- F. Morgan, Manifolds with Density and Perelman's Proof of the Poincare Conjecture, Am. Math. Mon., 116(2), (2009), 134-142. https://doi.org/10.1080/00029890.2009.11920920
- B. O'Neill, Elementary Differential Geometry, Academic Press Inc, (1966).
- B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, (1983).
- B. Ravani and J.W. Wang, Computer aided geometric design of line constructs, ASME J. Mech. Des., 113(4), (1991), 363-371. https://doi.org/10.1115/1.2912791
- C. Rosales, A. Canete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differ Equ., 31, (2008), 27-46. https://doi.org/10.1007/s00526-007-0104-y
- J.W. Rutter, Geometry of Curves, 1st. New York: Chapman and Hall/CRC; (2000), 384.
- S. Senyurt, Y. Altun and C. Cevahir, Smarandache curves for spherical indicatrix of the Bertrand curves pair, Boletim da Sociedade Paranaense de Matematica, 38(2), (2020), In Press, 27-39.
- M. Turgut and S. Yilmaz, Smarandache Curves in Minkowski Space-time, Int. J. Math. Comb., 3, (2008), 51-55.
- D.W. Yoon, Weighted minimal translation surfaces in Minkowski 3-space with density, Int. J. Geom. Methods Mod. Phys., 14, (2017), 175-178.
- D.W. Yoon, D.S. Kim, Y.H. Kim and J.W. Lee, Constructions of Helicoidal Surfaces in Euclidean Space with Density, Symmetry, (2017), 9, 173. https://doi.org/10.3390/sym9090173