scholarly journals Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

2020 ◽  
Vol 293 (4) ◽  
pp. 735-753 ◽  
Author(s):  
Rafael López ◽  
Álvaro Pámpano
Keyword(s):  
Euclidean Space ◽  
Linear Relation ◽  
Principal Curvatures ◽  
Surfaces In Euclidean Space ◽  
Rotational Surfaces

scholarly journals On Ribaucour transformations and applications to linear Weingarten surfaces

2002 ◽  
Vol 74 (4) ◽  
pp. 559-575 ◽  
Author(s):  
KETI TENENBLAT
Keyword(s):  
Euclidean Space ◽  
Principal Curvatures ◽  
Geometric Aspect ◽  
Space Forms ◽  
Flat Normal Bundle ◽  
Weingarten Surfaces ◽  
Ribaucour Transformations ◽  
Lines Of Curvature ◽  
Surfaces In Euclidean Space ◽  
Definition Of

We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.

scholarly journals On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

2020 ◽  
Vol 51 (4) ◽  
pp. 313-332
Author(s):  
Firooz Pashaie
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Principal Curvatures ◽  
First Variation ◽  
Chen’S Conjecture ◽  
The First Variation ◽  
Linearized Operator ◽  
Euclidean Submanifolds

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

The Development of Topological 4-manifold Theory

2021 ◽  
pp. 295-330
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Smooth Structures ◽  
Normal Bundles ◽  
Exotic Smooth Structures ◽  
Poincare Conjecture ◽  
Manifold Theory ◽  
Simply Connected

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

SUBMANIFOLDS WITH CONSTANT PRINCIPAL CURVATURES AND NORMAL HOLONOMY GROUPS

1991 ◽  
Vol 02 (02) ◽  
pp. 167-175 ◽  
Author(s):  
E. HEINTZE ◽  
C. OLMOS ◽  
G. THORBERGSSON
Keyword(s):  
Euclidean Space ◽  
Weyl Group ◽  
Principal Curvatures ◽  
Normal Field ◽  
Slice Theorem ◽  
Holonomy Groups ◽  
Shape Operators ◽  
Isoparametric Submanifold ◽  
Constant Principal Curvatures

A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators Aξ are constant for any parallel normal field ξ along any curve. It is shown that a submanifold of Euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold. Furthermore a "homogeneous slice theorem" is proved which says that the fibres of the projection of an isoparametric submanifold onto a full focal manifold are homogeneous isoparametric. To this end it is shown that any two points of an isoparametric submanifold can be connected by a piecewise differentiable curve whose pieces are tangent to one of the simultaneous eigenspaces Ei, i ∈ I, of the shape operators provided that the corresponding reflections generate the Weyl group of the isoparametric submanifold.

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