scholarly journals Twisted surfaces with vanishing curvature in Galilean 3-space

2017 ◽  
Vol 15 (01) ◽  
pp. 1850001 ◽  
Author(s):  
Mustafa Dede ◽  
Cumali Ekici ◽  
Wendy Goemans ◽  
Yasin Ünlütürk
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Planar Curve ◽  
Zero Gaussian Curvature ◽  
Zero Mean Curvature

In this work, we define twisted surfaces in Galilean 3-space. In order to construct these surfaces, a planar curve is subjected to two simultaneous rotations, possibly with different rotation speeds. The existence of Euclidean rotations and isotropic rotations leads to three distinct types of twisted surfaces in Galilean 3-space. Then we classify twisted surfaces in Galilean 3-space with zero Gaussian curvature or zero mean curvature.

scholarly journals COMPLETE STATIONARY SURFACES IN ${\mathbb R}^4_1$ WITH TOTAL GAUSSIAN CURVATURE – ∫ KdM = 4π

2013 ◽  
Vol 24 (11) ◽  
pp. 1350088
Author(s):  
XIANG MA ◽  
PENG WANG
Keyword(s):  
Lower Bound ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Lorentz Space ◽  
Double Covering ◽  
Möbius Strip ◽  
New Class ◽  
Stationary Surface ◽  
Zero Mean Curvature ◽  
Orientable Surfaces

We classify complete, algebraic, spacelike stationary (i.e. zero mean curvature) surfaces in four-dimensional Lorentz space [Formula: see text] with total Gaussian curvature – ∫ K d M = 4π. Such surfaces must be orientable surfaces, congruent to either the generalized catenoids or the generalized Enneper surfaces. The least total Gaussian curvature of a non-orientable algebraic stationary surface is 6π, which can be realized by Meeks' Möbius strip and its deformations (and also by a new class of non-algebraic examples). When the genus of its oriented double covering [Formula: see text] is g, we obtain the lower bound 2(g + 3)π, which is conjectured to be the best lower bound for each g.

A representation of distributions supported on smooth hypersurfaces of Rn

1995 ◽  
Vol 117 (1) ◽  
pp. 153-160
Author(s):  
Kanghui Guo
Keyword(s):  
Gaussian Curvature ◽  
Dual Space ◽  
Smooth Hypersurface ◽  
Zero Gaussian Curvature ◽  
Schwartz Class ◽  
Image Position

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

Limit equilibrium of reinforced shells of zero gaussian curvature

1989 ◽  
Vol 25 (9) ◽  
pp. 913-919
Author(s):  
A. V. Nalimov ◽  
Yu. V. Nemirovskii
Keyword(s):  
Gaussian Curvature ◽  
Limit Equilibrium ◽  
Zero Gaussian Curvature

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).

scholarly journals Modified Roller Coaster Surface in Space

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 195 ◽  
Author(s):  
Selçuk BAŞ ◽  
Talat KÖRPINAR
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Roller Coaster ◽  
Orthogonal Frame

In this paper, a new modified roller coaster surface according to a modified orthogonal frame is investigated in Euclidean 3-space. In this method, a new modified roller coaster surface is modeled. Both the Gaussian curvature and mean curvature of roller coaster surfaces are investigated. Subsequently, we obtain several characterizations in Euclidean 3-space.

Sign in / Sign up

Export Citation Format

Share Document