On Lorentzian surfaces in ℝ2,2

2017 ◽  
Vol 147 (1) ◽  
pp. 61-88
Author(s):  
Pierre Bayard ◽  
Victor Patty ◽  
Federico Sánchez-Bringas
Keyword(s):  
Fundamental Form ◽  
Gauss Map ◽  
Second Fundamental Form ◽  
Second Order ◽  
Lorentzian Surface ◽  
The Mean ◽  
Asymptotic Lines

We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

Geometric invariants and principal configurations on spacelike surfaces immersed in ℝ3,1

2010 ◽  
Vol 140 (6) ◽  
pp. 1141-1160 ◽  
Author(s):  
Pierre Bayard ◽  
Federico Sánchez-Bringas
Keyword(s):  
Second Fundamental Form ◽  
Spacelike Surface ◽  
Geometric Invariants ◽  
Dimensional Minkowski Space ◽  
Numerical Invariants ◽  
Curvature Ellipse ◽  
Curvature Lines ◽  
The Mean ◽  
Asymptotic Lines ◽  
Spacelike Surfaces

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

scholarly journals QUANTITATIVE OSCILLATION ESTIMATES FOR ALMOST-UMBILICAL CLOSED HYPERSURFACES IN EUCLIDEAN SPACE

2015 ◽  
Vol 92 (1) ◽  
pp. 133-144 ◽  
Author(s):  
JULIAN SCHEUER
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Uniformly Convex ◽  
Explicit Dependence ◽  
Closed Hypersurfaces ◽  
Bounded Volume ◽  
The Mean ◽  
Convex Hypersurfaces

We prove${\it\epsilon}$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is${\it\delta}$-small compared to the mean curvature. We give the explicit dependence of${\it\delta}$on${\it\epsilon}$within the class of uniformly convex hypersurfaces with bounded volume.

A note on rigidity of spacelike self-shrinkers

2020 ◽  
Vol 31 (05) ◽  
pp. 2050035
Author(s):  
Yong Luo ◽  
Hongbing Qiu
Keyword(s):  
Euclidean Space ◽  
Growth Condition ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Integral Method ◽  
Acta Math ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
A Minor

By using the integral method, we prove a rigidity theorem for spacelike self-shrinkers in pseudo-Euclidean space under a minor growth condition in terms of the mean curvature and the second fundamental form, which generalizes Theorem 1.1 in [H. Q. Liu and Y. L. Xin, Some Results on Space-Like Self-Shrinkers, Acta Math. Sin. (Engl. Ser.) 32(1) (2016) 69–82].

scholarly journals Self-Expanders of the Mean Curvature Flow

2021 ◽  
Author(s):  
Knut Smoczyk
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Vector ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Mean Convex

AbstractWe study self-expanding solutions $M^{m}\subset \mathbb {R}^{n}$ M m ⊂ ℝ n of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

scholarly journals Characterizations of the sphere by the mean II-curvature

1981 ◽  
Vol 23 (2) ◽  
pp. 249-253 ◽  
Author(s):  
George Stamou
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gaussian Curvature ◽  
Convex Surface ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Second Fundamental Form ◽  
The Mean

The notion of “mean II-curvature” of a C4-surface (without parabolic points) in the three-dimensional Euclidean space has been introduced by Ekkehart Glässner. The aim of this note is to give some global characterizations of the sphere related to the above notion.In the three-dimensional Euclidean space E3 we consider a sufficiently smooth ovaloid S (closed convex surface) with Gaussian curvature K > 0 . The ovaloid S possesses a positive definite second fundamental form II, if appropriately oriented. During the last years several authors have been concerned with the problem of characterizations of the sphere by the curvature of the second fundamental form of S. In this paper we give some characterizations of the sphere using the concept of the mean II-curvatureHII (of S), defined by Ekkehart Glässner.

scholarly journals Local isometric immersions of pseudo-spherical surfaces and kth order evolution equations

2019 ◽  
Vol 21 (04) ◽  
pp. 1850025
Author(s):  
Nabil Kahouadji ◽  
Niky Kamran ◽  
Keti Tenenblat
Keyword(s):  
Isometric Immersion ◽  
Evolution Equations ◽  
Second Fundamental Form ◽  
Second Order ◽  
Isometric Immersions ◽  
Spherical Surfaces ◽  
Pseudospherical Surfaces ◽  
The Mean ◽  
Local Isometric Immersion ◽  
Straight Lines

We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local isometric immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local isometric immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an isometric immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local isometric immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the isometric immersion.

scholarly journals Twisted product CR-submanifolds in a locally conformal Kähler space forms

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1915-1925
Author(s):  
Vittoria Bonanzinga ◽  
Koji Matsumoto
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Twisted Product ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Locally Conformal Kähler ◽  
The Mean ◽  
Locally Conformal ◽  
Cr Submanifolds

Certain twisted product CR-submanifolds in a K?hler manifold and some inequalities of the second fundamental form of these submanifolds are presented ([14]). Then the length of the second fundamental form of a twisted product CR-submanifold in a locally conformal K?hler manifold is considered (2013), ([15]). In this paper, we consider the relation of the mean curvature and the length of the second fundamental form in two twisted product CR-submanifolds in a locally conformal K?hler space forms.

scholarly journals A gauge theory of nucleonic interactions by contact

2014 ◽  
Vol 29 (14) ◽  
pp. 1450073 ◽  
Author(s):  
Nicolae Mazilu ◽  
Pavlos D. Ioannou ◽  
Maricel Agop
Keyword(s):  
Gauge Theory ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Local Deformation ◽  
Riemann Space ◽  
General Idea ◽  
The Second Fundamental Form ◽  
The Mean ◽  
Definition Of ◽  
Contact Spots

A gauge theory of contact is presented, based on the general idea that the local deformation of the nucleon surface at contact should be gauged by the variation of curvature. A contact force is then defined so as to cope with both the variation of curvature and the deformation. This force generalizes the classical definition of surface tension, in that it depends on the mean curvature, but also depends on the variance of the second fundamental form of surface, considered as a statistical variable over the ensemble of contact spots. It turns out that the variance of the second fundamental form does not depend but on the metric of the space of curvature parameters, organized as Riemann space. This result compels us to review the definition of physical surface of a nucleon.

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