scholarly journals CONSTANT SCALAR CURVATURE HYPERSURFACES WITH SECOND-ORDER UMBILICITY

2009 ◽  
Vol 51 (2) ◽  
pp. 219-241
Author(s):  
ANTONIO GERVASIO COLARES ◽  
FERNANDO ENRIQUE ECHAIZ-ESPINOZA
Keyword(s):  
Isometric Immersion ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Einstein Manifolds ◽  
Isometric Immersions ◽  
Principal Curvatures ◽  
Space Forms ◽  
Newton Polynomial ◽  
Product Of Spheres ◽  
The Mean

AbstractWe extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres $S^{k}(\frac{1}{\sqrt{2}})\times S^{k}(\frac{1}{\sqrt{2}})$ embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions

2009 ◽  
Vol 61 (3) ◽  
pp. 641-655
Author(s):  
Sadahiro Maeda ◽  
Seiichi Udagawa
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
Curvature Vector ◽  
Second Fundamental Form ◽  
Isometric Immersions ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
The Mean

Abstract.For an isotropic submanifold Mn (n ≧ 3) of a space form of constant sectional curvature c, we show that if the mean curvature vector of Mn is parallel and the sectional curvature K of Mn satisfies some inequality, then the second fundamental form of Mn in is parallel and our manifold Mn is a space form.

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 Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

1972 ◽  
Vol 45 ◽  
pp. 139-165 ◽  
Author(s):  
Joseph Erbacher
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Additional Assumption ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Normal Connection ◽  
Isometric Immersions ◽  
The Mean ◽  
Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

HYPERSURFACES IN A UNIT SPHERE Sn+1(1) WITH CONSTANT SCALAR CURVATURE

2001 ◽  
Vol 64 (3) ◽  
pp. 755-768 ◽  
Author(s):  
QING-MING CHENG
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Conformally Flat ◽  
Principal Curvatures ◽  
The Second Fundamental Form ◽  
Locally Conformally Flat ◽  
Complete Hypersurfaces ◽  
Conformally Flat Hypersurface

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and(1) when r ≠ (n−2)/(n−1), ifthen M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

Complete spacelike hypersurfaces in a de Sitter space

2006 ◽  
Vol 73 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Shu Shichang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
The Mean

In this paper, we characterise the n-dimensional (n ≥ 3) complete spacelike hypersurfaces Mn in a de Sitter space with constant scalar curvature and with two distinct principal curvatures. We show that if the multiplicities of such principal curvatures are greater than 1, then Mn is isometric to Hk (sinh r) × Sn−k (cosh r), 1 < k < n − 1. In particular, when Mn is the complete spacelike hypersurfaces in with the scalar curvature and the mean curvature being linearly related, we also obtain a characteristic Theorem of such hypersurfaces.

ON COMPLETE SPACELIKE HYPERSURFACES WITH CONSTANT m-TH MEAN CURVATURE IN AN ANTI-DE SITTER SPACE

2010 ◽  
Vol 21 (05) ◽  
pp. 551-569 ◽  
Author(s):  
B. Y. WU
Keyword(s):  
Mean Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
De Sitter ◽  
Principal Curvatures ◽  
Global Rigidity ◽  
Complete Spacelike Hypersurfaces ◽  
Spacelike Hypersurfaces ◽  
Hyperbolic Cylinders

We investigate complete spacelike hypersurfaces in an Anti-de Sitter space with constant m-th mean curvature and two distinct principal curvatures. By using Otsuki's idea, we obtain some global classification results. For their application, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Anti-de Sitter (n + 1)-spaces (n ≥ 3) of constant mean curvature or constant scalar curvature with two distinct principal curvatures λ and μ satisfying inf (λ - μ)2 > 0 are the hyperbolic cylinders. It is a little surprising that the corresponding result does not hold for m-th mean curvature when m > 2. We also obtain some global rigidity results for hyperbolic cylinders in terms of square length of the second fundamental form.

scholarly journals A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1238
Author(s):  
Pablo Alegre ◽  
Joaquín Barrera ◽  
Alfonso Carriazo
Keyword(s):  
Closed Form ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms ◽  
Maslov Form ◽  
The Mean

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

scholarly journals Mean curvature and shape operator of isometric immersions in real-space-forms

1996 ◽  
Vol 38 (1) ◽  
pp. 87-97 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Shape Operator ◽  
Real Space ◽  
Curvature Function ◽  
Isometric Immersions ◽  
Space Forms ◽  
Real Space Forms ◽  
The Mean

According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

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 Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Tongzhu Li ◽  
Changxiong Nie
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Lorentz Space ◽  
Constant Mean Curvature ◽  
Conformal Geometry ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Lagrange Equations ◽  
Conformal Metric ◽  
Space Forms

Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in .

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