scholarly journals Laguerre hypersurfaces with definite Laguerre tensor

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2099-2106
Author(s):  
Jianbo Fang ◽  
Fengjiang Li ◽  
Jianxiang Li
Keyword(s):  
Fundamental Form ◽  
Transformation Group ◽  
Second Fundamental Form ◽  
Principal Curvatures ◽  
Laguerre Transformation ◽  
Laguerre Tensor ◽  
Laguerre Second Fundamental Form ◽  
Laguerre Transformation Group

x : Mn-1 ? Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. In this paper, we will study the Laguerre hypersurface with parallel Laguerre form and nonnegative or non-positive Laguerre tensor.

scholarly journals Curves and surfaces in the context of optometry. Part 2: Surfaces

2006 ◽  
Vol 65 (1) ◽  
Author(s):  
W.F. Harris
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Surface Curvature ◽  
Principal Curvatures ◽  
The Second Fundamental Form ◽  
Curves And Surfaces ◽  
Length And Area ◽  
First Fundamental Form ◽  
Differential Geom

This paper introduces the differential geom-etry of surfaces in Euclidean 3-space. The first and second fundamental forms of a surface are defined.   The  first  fundamental  form  provides a metric for calculations of length and area on the surface. The second fundamental form deter-mines surface curvature and, hence, concepts of importance in optometry such as surface power and sagitta. The principal curvatures at a point on a surface are obtained as solutions of a qua-dratic equation. The torus is used to illustrate the methods.

scholarly journals PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

2012 ◽  
Vol 54 (3) ◽  
pp. 579-597 ◽  
Author(s):  
SHICHANG SHU ◽  
BIANPING SU
Keyword(s):  
Fundamental Form ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Isoparametric Hypersurfaces ◽  
Blaschke Tensor ◽  
Blaschke Isoparametric Hypersurface ◽  
Full Classification ◽  
Möbius Second Fundamental Form

AbstractLet A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

scholarly journals Hypersurfaces with constant mean curvature and two principal curvatures in n+1

2004 ◽  
Vol 76 (3) ◽  
pp. 489-497 ◽  
Author(s):  
Luis J. Alías ◽  
Sebastião C. de Almeida ◽  
Aldir Brasil Jr.
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Integral Formula ◽  
Second Fundamental Form ◽  
Euclidean Sphere ◽  
Principal Curvatures ◽  
Clifford Torus ◽  
The Second Fundamental Form

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

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