Curvature Vector

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The Experts below are selected from a list of 351 Experts worldwide ranked by ideXlab platform

Miran Saje - One of the best experts on this subject based on the ideXlab platform.

Velichka Milousheva - One of the best experts on this subject based on the ideXlab platform.

  • surfaces with parallel normalized mean Curvature Vector field in euclidean or minkowski 4 space
    Filomat, 2019
    Co-Authors: Georgi Ganchev, Velichka Milousheva
    Abstract:

    We study surfaces with parallel normalized mean Curvature Vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean Curvature Vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean Curvature Vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

  • on the theory of lorentz surfaces with parallel normalized mean Curvature Vector field in pseudo euclidean 4 space
    Journal of The Korean Mathematical Society, 2016
    Co-Authors: Yana Aleksieva, Georgi Ganchev, Velichka Milousheva
    Abstract:

    We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean Curvature Vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.

  • quasi minimal rotational surfaces in pseudo euclidean four dimensional space
    Open Mathematics, 2014
    Co-Authors: Georgi Ganchev, Velichka Milousheva
    Abstract:

    In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis — rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean Curvature Vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.

  • an invariant theory of marginally trapped surfaces in the four dimensional minkowski space
    Journal of Mathematical Physics, 2012
    Co-Authors: Georgi Ganchev, Velichka Milousheva
    Abstract:

    A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean Curvature Vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meridian surfaces.

Bang-yen Chen - One of the best experts on this subject based on the ideXlab platform.

  • chen s biharmonic conjecture and submanifolds with parallel normalized mean Curvature Vector
    Mathematics, 2019
    Co-Authors: Bang-yen Chen
    Abstract:

    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 parallel mean Curvature Vector in riemannian and indefinite space forms
    arXiv: Differential Geometry, 2013
    Co-Authors: Bang-yen Chen
    Abstract:

    A submanifold of a pseudo-Riemannian manifold is said to have parallel mean Curvature Vector if the mean Curvature Vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean Curvature Vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on submanifolds with parallel mean Curvature Vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel mean Curvature Vector in Riemannian and indefinite space forms.

  • complete classification of lorentz surfaces with parallel mean Curvature Vector in arbitrary pseudo euclidean space
    Kyushu Journal of Mathematics, 2010
    Co-Authors: Bang-yen Chen
    Abstract:

    Surfaces with parallel mean Curvature Vector play important roles in the theory of harmonic maps, differential geometry as well as in physics. Surfaces with parallel mean Curvature Vector in Riemannian space forms were classified in the early 1970s by Chen and Yau. Recently, space-like surfaces with parallel mean Curvature Vector in arbitrary indefinite space forms were completely classified by Chen in two papers in 2009. In this paper, we completely classify Lorentz surfaces with parallel mean Curvature Vector in a pseudo-Euclidean space Ems with arbitrary dimension m and arbitrary index s. Our main result states that there are 23 families of Lorentz surfaces with parallel mean Curvature Vector in a pseudo-Euclidean m-space Ems . Conversely, every Lorentz surface with parallel mean Curvature Vector in Ems is obtained from the 23 families.

  • Complete classification of spatial surfaces with parallel mean Curvature Vector in arbitrary non-flat pseudo-Riemannian space forms
    Central European Journal of Mathematics, 2009
    Co-Authors: Bang-yen Chen
    Abstract:

    Submanifolds with parallel mean Curvature Vector play important roles in differential geometry, theory of harmonic maps as well as in physics. Spatial surfaces in 4D Lorentzian space forms with parallel mean Curvature Vector were classified by B. Y. Chen and J. Van der Veken in [9]. Recently, spatial surfaces with parallel mean Curvature Vector in arbitrary pseudo-Euclidean spaces are also classified in [7]. In this article, we classify spatial surfaces with parallel mean Curvature Vector in pseudo-Riemannian spheres and pseudo-hyperbolic spaces with arbitrary codimension and arbitrary index. Consequently, we achieve the complete classification of spatial surfaces with parallel mean Curvature Vector in all pseudo-Riemannian space forms. As an immediate by-product, we obtain the complete classifications of spatial surfaces with parallel mean Curvature Vector in arbitrary Lorentzian space forms.

  • classification of quasi minimal surfaces with parallel mean Curvature Vector in pseudo euclidean 4 space mathbb e 4_2
    Results in Mathematics, 2009
    Co-Authors: Bang-yen Chen, Oscar J Garay
    Abstract:

    A surface in the pseudo-Euclidean space $${\mathbb E}^4_2$$ with neutral metric (or in the Lorentzian complex plane C12) is called quasi-minimal if its mean Curvature Vector is light-like at each point. Such surface are always Lorentzian. In this article, we completely classify quasi-minimal surfaces with parallel mean Curvature Vector in the pseudo-Euclidean space $${\mathbb E}^4_2$$ .

Uğur Dursun - One of the best experts on this subject based on the ideXlab platform.

  • pseudo spherical submanifolds with 1 type pseudo spherical gauss map
    Results in Mathematics, 2017
    Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur Dursun
    Abstract:

    In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere \({\mathbb{S}^4_s(1)}\) with index s, \({s=1, 2}\), and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere \({\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s}\) with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space \({\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1}\) with 1-type pseudo-spherical Gauss map. Finally, according to the causal character of the mean Curvature Vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant component in its spectral decomposition.

  • pseudo spherical submanifolds with 1 type pseudo spherical gauss map
    arXiv: Differential Geometry, 2015
    Co-Authors: Burcu Bektaş, Elif Özkara Canfes, Uğur Dursun
    Abstract:

    In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type pseudo-spherical Gauss map. Finally, according to the causal character of the mean Curvature Vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant component in its spectral decomposition.

  • null 2 type submanifolds of the euclidean space e 5 with non parallel mean Curvature Vector
    Journal of Geometry, 2007
    Co-Authors: Uğur Dursun
    Abstract:

    Let M be a 3-dimensional submanifold of the Euclidean space E5 such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean Curvature and non-parallel mean Curvature Vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean Curvature and non-parallel mean Curvature Vector.

  • null 2 type submanifolds of the euclidean space e5 with parallel normalized mean Curvature Vector
    Kodai Mathematical Journal, 2005
    Co-Authors: Uğur Dursun
    Abstract:

    We classify 3-dimensional null 2-type submanifolds of the Euclidean space E5 with parallel normalized mean Curvature Vector under certain hypothesis.

Dejan Zupan - One of the best experts on this subject based on the ideXlab platform.

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