Timelike tubular surfaces of Weingarten types and linear Weingarten types in Minkowski 3-space
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $K$, $H$, $K_{II}$ and $H_{II}$ be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface $T_\gamma(\alpha)$ with the radius $\gamma$ along a timelike curve $\alpha(s)$ in Minkowski 3-space $E_{1}^3$. We prove that $T_\gamma(\alpha)$ must be a $(K,H)$-Weingarten surface and a $(K,H)$-linear Weingarten surface. We also show that $T_{\gamma}(\alpha)$ is $(X,Y)$-Weingarten type if and only if its central curve is a circle or a helix, where $(X,Y)$ $\in$ $\{(K,K_{II})$, $(K,H_{II})$, $(H,K_{II})$, $(H,H_{II})$, $(K_{II}$, $H_{II}) \}$. Furthermore, we prove that there exist no timelike tubular surfaces of $(X,Y)$-linear Weingarten type, $(X,Y,Z)$-linear Weingarten type and $(K,H,K_{II},H_{II})$-linear Weingarten type along a timelike curve in $E_{1}^3$, where $(X,Y,Z)\in\{(K,H,K_{II})$, $(K,H,H_{II})$, $(K,K_{II},H_{II})$, $(H$, $K_{II},H_{II})\}$.
Abstract
Let $K$, $H$, $K_{II}$ and $H_{II}$ be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface $T_\gamma(\alpha)$ with the radius $\gamma$ along a timelike curve $\alpha(s)$ in Minkowski 3-space $E_{1}^3$. We prove that $T_\gamma(\alpha)$ must be a $(K,H)$-Weingarten surface and a $(K,H)$-linear Weingarten surface. We also show that $T_{\gamma}(\alpha)$ is $(X,Y)$-Weingarten type if and only if its central curve is a circle or a helix, where $(X,Y)$ $\in$ $\{(K,K_{II})$, $(K,H_{II})$, $(H,K_{II})$, $(H,H_{II})$, $(K_{II}$, $H_{II}) \}$. Furthermore, we prove that there exist no timelike tubular surfaces of $(X,Y)$-linear Weingarten type, $(X,Y,Z)$-linear Weingarten type and $(K,H,K_{II},H_{II})$-linear Weingarten type along a timelike curve in $E_{1}^3$, where $(X,Y,Z)\in\{(K,H,K_{II})$, $(K,H,H_{II})$, $(K,K_{II},H_{II})$, $(H$, $K_{II},H_{II})\}$.
참고문헌 (22)
- B. Brunt, "Potential applications of Weingarten surfaces in CAGD. Part I : Weingarten surfaces and surface shape investigation", Comput. Aided Geom. Design, 13(6), pp.569~582 (1996)[DOI]
- S. S. Chern, "Some new characterizations of the Euclidean sphere", Duke Math. J, 12, pp.279~290 (1945)
- F. Dillen, "Ruled Weingarten surfaces in Minkowski 3-space", Manuscripta Math, 98(3), pp.307~320 (1999)[DOI]
- F. Dillen, "Ruled surfaces of Weingarten type in Minkowski 3-space", J. Geom, 83(1-2), pp.10~21 (2005)[DOI]
- X. Fu, "Geometric characterizations of canal surfaces in Minkowski 3-space I", Bull. Korean Math. Soc, 56(4), pp.867~883 (2019)[DOI]
- J. A. Galvez, "Linear Weingarten surfaces in R3", Monatsh. Math, 138(2), pp.133~144 (2003)[DOI]
- H. Hopf, "Uber Flachen mit einer Relation zwischen den Hauptkrummungen", Math. Nachr, 4, pp.232~249 (1951)[DOI]
- M. K. Karacan, "Tubular surfaces of Weingarten types in Minkowski 3-space", Gen. Math. Notes, 22(1), pp.44~56 (2014)
- Y. H. Kim, "Classification of ruled surfaces in Minkowski 3-spaces", J. Geom. Phys, 49(1), pp.89~100 (2004)[DOI]
- W. Kuhnel, "On closed Weingarten surfaces", Monatsh. Math, 146(2), pp.113~126 (2005)[DOI]
- R. Lopez, "On linear Weingarten surfaces", Internat. J. Math, 19(4), pp.439~448 (2008)[DOI]
- R. Lopez, "Differential geometry of curves and surfaces in Lorentz-Minkowski space", Int. Electron. J. Geom, 7(1), pp.44~107 (2014)
- R. Lopez, "Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures", Math. Nachr, 293(4), pp.735~753 (2020)[DOI]
- M. G. Puopolo, "Velocity control of a cylindrical rolling robot by shape changing", Adv. Robotics, 30(23), pp.1484~1494 (2016)
- J. H. Qian, "Geometric characterizations of canal surfaces in Minkowski 3-space II", Math, 7(8), pp.703~ (2019)
- J. S. Ro, "Tubes of weingarten types in a Euclidean 3-space", J. Chungcheong Math. Soc, 22(3), pp.359~366 (2009)
- H. Rosenberg, "The geometry of properly embedded special surfaces in R3; e.g., surfaces satisfying aH + bK = 1, where a and b are positive", Duke Math. J, 73(2), pp.291~306 (1994)[DOI]
- Y. Tuncer, "Weingarten and linear Weingarten type tubular surfaces in E3", Math. Probl. Eng, pp.191849~ (2011)[DOI]
- A. U¸cum, "New types of canal surfaces in Minkowski 3-space", Adv. Appl. Clifford Algebr, 26(1), pp.449~468 (2016)[DOI]
- K. Voss, "Uber geschlossene Weingartensche Flachen", Math. Ann, 138, pp.42~54 (1959)[DOI]
- J. Weingarten, "Ueber eine Klasse auf einander abwickelbarer Fl¨achen", J. Reine Angew. Math, 59, pp.382~393 (1861)[DOI]
- J. Weingarten, "Ueber die Fl¨achen deren Normalen eine gegebene Fl¨ache ber¨uhren", J. Reine Angew. Math, 62, pp.61~63 (1863)[DOI]