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.

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.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

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.

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.

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