scholarly journals Some Characterizations of Generalized Null Scrolls

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 931 ◽  
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Seoung Dal Jung
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Structure Functions ◽  
Ruled Surfaces ◽  
Base Curve ◽  
Second Gaussian Curvature

In this work, a family of ruled surfaces named generalized null scrolls in Minkowski 3-space are investigated via the defined structure functions. The relations between the base curve and the ruling flow of the generalized null scroll are revealed. The Gaussian curvature, mean curvature, second Gaussian curvature and the second mean curvature are given and related to each other. Last but not least, the generalized null scrolls whose base curves are k-type null helices are discussed and several examples are presented.

scholarly journals On the second Gaussian curvature of ruled surfaces in Euclidean $3$-space

2006 ◽  
Vol 37 (3) ◽  
pp. 221-226 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Mean ◽  
Second Gaussian Curvature

In this paper, we mainly investigate non developable ruled surface in a 3-dimensional Euclidean space satisfying the equation $K_{II} = KH$ along each ruling, where $K$ is the Gaussian curvature, $H$ is the mean curvature and $K_{II}$ is the second Gaussian curvature.

scholarly journals Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

2020 ◽  
Vol 28 (3) ◽  
pp. 229-240
Author(s):  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
First Order ◽  
Order Nonlinear Differential Equation ◽  
Base Curve

AbstractIn Lorentz-Minkowski 3-space, null scrolls are ruled surfaces with a null base curve and null rulings. Their mean, as well as their Gaussian curvature, depends only on a parameter of a base curve. In the present paper, we obtain the first-order nonlinear differential equation (Riccati equation) which relates curvatures of a base curve to curvatures of a null scroll. Conditioned by this equation, we can determine a family of null scrolls with a given null base curve and prescribed curvatures, in particular, a family of minimal and constant mean curvature null scrolls.

Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space

2019 ◽  
Vol 16 (05) ◽  
pp. 1950076 ◽  
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
Null Curve ◽  
Base Curve

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

scholarly journals Weingarten and Linear Weingarten Type Tubular Surfaces in

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Yılmaz Tunçer ◽  
Dae Won Yoon ◽  
Murat Kemal Karacan
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
The Mean ◽  
Second Gaussian Curvature ◽  
Tubular Surfaces

We study tubular surfaces in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature, and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Euclidean 3-space.

scholarly journals Offset Ruled Surface in Euclidean Space with Density

2021 ◽  
Vol 29 (1) ◽  
pp. 219-233
Author(s):  
Neslihan Ulucan ◽  
Mahmut Akyigit
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Surfaces In Euclidean Space ◽  
The Mean

Abstract In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

Ruled surfaces with vanishing second Gaussian curvature

1992 ◽  
Vol 113 (3) ◽  
pp. 177-181 ◽  
Author(s):  
D. E. Blair ◽  
Th. Koufogiorgos
Keyword(s):  
Gaussian Curvature ◽  
Ruled Surfaces ◽  
Second Gaussian Curvature

Non-lightlike ruled surfaces with constant curvatures in Minkowski 3-space

2018 ◽  
Vol 15 (04) ◽  
pp. 1850068 ◽  
Author(s):  
Ahmad Tawfik Ali
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surface ◽  
Timelike Curve ◽  
Ruled Surfaces ◽  
Base Curve ◽  
Timelike Ruled Surface ◽  
Interesting Theorem

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] are the tangent, principal normal and binormal vectors of an arbitrary timelike curve [Formula: see text]. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.

scholarly journals RULED SURFACES AND TANGENT BUNDLE OF PSEUDO-SPHERE OF NATURAL LIFT CURVES

2020 ◽  
Vol 20 (3) ◽  
pp. 573-586
Author(s):  
EMEL KARACA ◽  
MUSTAFA CALISKAN
Keyword(s):  
Unit Sphere ◽  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Mean Curvature ◽  
Shape Operator ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Curve Shape ◽  
Lift Curve

In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, TM, and Lorentzian unit sphere, S12 is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, TM, and hyperbolic unit sphere, H2 is given. According to E. Study mapping, any curve on S12 or H2 corresponds to a ruled surface in R13. By constructing these isomorphisms, we correspond to any natural lift curve on TM or TM a unique ruled surface in R13. Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.

scholarly journals Umbilical points on surfaces in RN

1985 ◽  
Vol 100 ◽  
pp. 135-143 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Normal Connection ◽  
Round Sphere ◽  
Positive Gaussian Curvature ◽  
Connected Surface ◽  
Umbilical Points ◽  
The Mean

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

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