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Dae Ho Jin - One of the best experts on this subject based on the ideXlab platform.
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non existence for screen quasi conformal irrotational half lightlike submanifolds of a semi Riemannian Space form admitting a semi symmetric non metric connection
East Asian mathematical journal, 2015Co-Authors: Dae Ho JinAbstract:We study screen quasi-conformal irrotational half lightlike submanifolds M of a semi-Riemannian Space form (c) equipped with a semi-symmetric non-metric connection subject such that the structure vector field of (c) belongs to the screen distribution S(TM). The main result is a non-existence theorem for such half lightlike submanifolds.
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ascreen lightlike hypersurfaces of a semi Riemannian Space form with a semi symmetric non metric connection
Communications of The Korean Mathematical Society, 2014Co-Authors: Dae Ho JinAbstract:Abstract. We study lightlike hypersurfaces of a semi-Riemannian Spaceform fM(c) admitting a semi-symmetric non-metric connection. First, weconstruct a type of lightlike hypersurfaces according to the form of thestructure vector field of Mf(c), which is called a ascreen lightlike hyper-surface. Next, we prove a characterization theorem for such an ascreenlightlike hypersurface endow with a totally geodesic screen distribution. 1. IntroductionThe theory of lightlike submanifolds is an important topic of research in dif-ferential geometry due to its application in mathematical physics, especially inthe electromagneticfield theory. The study ofsuch notion wasinitiated by Dug-gal and Bejancu [3] and later studied by many authors (see up-to date resultsin two books [4, 5]). The notion of a semi-symmetric non-metric connectionon a Riemannian manifold was introduced by Ageshe and Chafle [1]. Recentlyseveral authors ([9]-[13]) studied lightlike hypersurfaces in a semi-Riemannianmanifold admitting a semi-symmetric non-metric connection. Most of authorsthat wrote on either lightlike hypersurfaces M of semi-Riemannian manifoldsMfadmitting semi-symmetric non-metric connections or lightlike hypersurfacesM of indefinite almost contact manifolds Mffail to treat with the case the struc-ture vector field ζ of Mfis not tangent to M, but studied only to the case ζis tangent to M (such M is called tangential lightlike submanifold ([9]-[13]) ofMf). There are few papers on non-tangential lightlike submanifolds of indefinitealmost contact manifolds studied by Jin ([6]-[8]).In this paper, we study non-tangential lightlike hypersurfaces of a semi-Riemannian Space form admitting a semi-symmetric non-metric connection.There are several different types of non-tangential lightlike hypersurfaces ac-cording to the form of the structure vector field of the ambient manifold. We
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half lightlike submanifolds of a semi Riemannian Space form with a semi symmetric non metric connection
Pure and Applied Mathematics, 2014Co-Authors: Dae Ho JinAbstract:In this paper, we study screen quasi-conformal irrotational half lightlike submanifolds M of a semi-Riemannian Space form f M(c) admitting a semi-symmetric non-metric connection, whose structure vector fleld ‡ is tangent to M. The main result is a classiflcation theorem for such Einstein half lightlike submanifolds of a Lorentzian Space form admitting a semi-symmetric non-metric connection.
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geometry of half lightlike submanifolds of a semi Riemannian Space form with a semi symmetric metric connection
Journal of the Chungcheong Mathematical Society, 2011Co-Authors: Dae Ho JinAbstract:We study the geometry of half lightlike sbmanifolds M of a semi-Riemannian Space form admitting a semi-symmetric metric connection subject to the conditions: (1) The screen distribution S(TM) is totally umbilical (geodesic) and (2) the co-screen distribution S() of M is a conformal Killing one.
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screen conformal lightlike hypersurfaces of a semi Riemannian Space form
Pure and Applied Mathematics, 2009Co-Authors: Dae Ho JinAbstract:We study the geometry of screen conformal light like hypersurfaces M of a semi- Riemannian manifold M. The main result is a characterization theorem for screen conformal lightlike hypersurfaces of a semi-Riemannian Space form.
Huili Liu - One of the best experts on this subject based on the ideXlab platform.
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curves in three dimensional Riemannian Space forms
Journal of Geometry, 2021Co-Authors: Huili Liu, Yixuan LiuAbstract:In the three dimensional Riemannian Space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean Space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic Space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on Space forms.
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curves in three dimensional Riemannian Space forms
Results in Mathematics, 2014Co-Authors: Huili LiuAbstract:In three dimensional Riemannian Space forms, introducing a natural moving frame, we define the associate curve of a curve and study the properties and relations of a curve and its associate curve. We state necessary and sufficient condition that a Frenet curve is a Bertrand curve in three dimensional Riemannian Space forms, especially in a Riemannian 3-dimensional sphere and in a 3-dimensional hyperbolic Space, resp. At the same time we give an explicit expression of the partner curve of a Bertrand curve.
Bang-yen Chen - One of the best experts on this subject based on the ideXlab platform.
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complete classification of lorentz surfaces with parallel mean curvature vector in arbitrary pseudo euclidean Space
Kyushu Journal of Mathematics, 2010Co-Authors: Bang-yen ChenAbstract: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.
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complete classification of parallel spatial surfaces in pseudo Riemannian Space forms with arbitrary index and dimension
Journal of Geometry and Physics, 2010Co-Authors: Bang-yen ChenAbstract:Abstract A spatial surface of a pseudo-Riemannian Space form is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. It is well known that a surface in a pseudo-Riemannian Space form is parallel if and only if it is locally invariant under the reflection with respect to the normal Space at each point. Such surfaces are important in geometry as well as in general relativity since the extrinsic invariants of the surfaces do not change from point to point. Recently, parallel spatial surfaces in 4-dimensional Lorentzian Space forms were classified by Chen and Van der Veken (2009) [6] . In this article, we completely classify parallel spatial surfaces in pseudo-Riemannian Space forms with an arbitrary index and dimensions. As an immediate by-product, we achieve the classification of all spatial surfaces in Lorentzian Space forms with arbitrary dimensions.
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Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian Space forms
Central European Journal of Mathematics, 2009Co-Authors: Bang-yen ChenAbstract: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.
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characterizations of Riemannian Space forms einstein Spaces and conformally flat Spaces
Proceedings of the American Mathematical Society, 1999Co-Authors: Bang-yen Chen, Leopold Verstraelen, Franki Dillen, Luc VranckenAbstract:In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold M, denoted respectively by 6(nl,... ,nk) and 5(ni,... ,nk), which trivially satisfy 6(ni,... ,nk) > 6(n,... , nk). In this article, we completely determine the Riemannian manifolds satisfying the condition 6(nfl,... , nk) = S(nl,... ,nk). By applying the notions of these 6-invariants, we establish new characterizations of Einstein and conformally flat Spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.
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relations between ricci curvature and shape operator for submanifolds with arbitrary codimensions
Glasgow Mathematical Journal, 1999Co-Authors: Bang-yen ChenAbstract:First we define the notion of k-Ricci curvature of a Riemannian n- manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean cur- vature for a submanifold in a Riemannian Space form with arbitrary codimension. Several applications of such relationships are also presented.
Julien Roth - One of the best experts on this subject based on the ideXlab platform.
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spinorial representation of submanifolds in Riemannian Space forms
Pacific Journal of Mathematics, 2017Co-Authors: Pierre Bayard, Marie-amélie Lawn, Julien RothAbstract:In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian Space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.
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Spinorial characterizations of surfaces into 3-dimensional psuedo-Riemannian Space forms
Mathematical Physics Analysis and Geometry, 2011Co-Authors: Marie-amélie Lawn, Julien RothAbstract:We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian Space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\mathbb{R}^{2,1}$ to other Lorentzian Space forms. We also characterize immersions of Riemannian surfaces in these Spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in Space forms of corresponding signature, as well as for Spacelike and timelike immersions of surfaces of signature (0,2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.
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Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms
Mathematical Physics Analysis and Geometry, 2011Co-Authors: Marie-amélie Lawn, Julien RothAbstract:We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian Space forms. This generalizes a recent work of the first author for Spacelike immersed Lorentzian surfaces in ℝ^2,1 to other Lorentzian Space forms. We also characterize immersions of Riemannian surfaces in these Spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in Space forms of corresponding signature, as well as for Spacelike and timelike immersions of surfaces of signature (0, 2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.
Martin Jagersand - One of the best experts on this subject based on the ideXlab platform.
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tumor invasion margin on the Riemannian Space of brain fibers
Medical Image Analysis, 2012Co-Authors: Parisa Mosayebi, Dana Cobzas, Albert Murtha, Martin JagersandAbstract:Abstract Glioma is one of the most challenging types of brain tumors to treat or control locally. One of the main problems is to determine which areas of the apparently normal brain contain glioma cells, as gliomas are known to infiltrate several centimeters beyond the clinically apparent lesion that is visualized on standard Computed Tomography scans (CT) or Magnetic Resonance Images (MRIs). To ensure that radiation treatment encompasses the whole tumor, including the cancerous cells not revealed by MRI, doctors treat the volume of brain that extends 2 cm out from the margin of the visible tumor. This approach does not consider varying tumor-growth dynamics in different brain tissues, thus it may result in killing some healthy cells while leaving cancerous cells alive in the other areas. These cells may cause recurrence of the tumor later in time, which limits the effectiveness of the therapy. Knowing that glioma cells preferentially spread along nerve fibers, we propose the use of a geodesic distance on the Riemannian manifold of brain diffusion tensors to replace the Euclidean distance used in the clinical practice and to correctly identify the tumor invasion margin. This mathematical model results in a first-order Partial Differential Equation (PDE) that can be numerically solved in a stable and consistent way. To compute the geodesic distance, we use actual Diffusion Weighted Imaging (DWI) data from 11 patients with glioma and compare our predicted infiltration distance map with actual grwoth in follow-up MRI scans. Results show improvement in predicting the invasion margin when using the geodesic distance as opposed to the 2 cm conventional Euclidean distance.
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tumor invasion margin on the Riemannian Space of brain fibers
Medical Image Computing and Computer-Assisted Intervention, 2009Co-Authors: Dana Cobzas, Parisa Mosayebi, Albert Murtha, Martin JagersandAbstract:Gliomas are one of the most challenging tumors to treat or control locally. One of the main challenges is determining which areas of the apparently normal brain contain glioma cells, as gliomas are known to infiltrate for several centimeters beyond the clinically apparent lesion visualized on standard CT or MRI. To ensure that radiation treatment encompasses the whole tumour, including the cancerous cells not revealed by MRI, doctors treat a volume of brain extending 2cm out from the margin of the visible tumour. This expanded volume often includes healthy, non-cancerous brain tissue. Knowing that glioma cells preferentially spread along nerve fibers, we propose the use of a geodesic distance on the Riemannian manifold of brain fibers to replace the Euclidean distance used in clinical practice and to correctly identify the tumor invasion margin. To compute the geodesic distance we use actual DTI data from patients with glioma and compare our predicted growth with follow-up MRI scans. Results show improvement in predicting the invasion margin when using the geodesic distance as opposed to the 2cm conventional Euclidean distance.
