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Keywords = isotropic submanifolds

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13 pages, 263 KiB  
Article
Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection
by Yanlin Li, Md Aquib, Meraj Ali Khan, Ibrahim Al-Dayel and Khalid Masood
Axioms 2024, 13(7), 454; https://doi.org/10.3390/axioms13070454 - 4 Jul 2024
Cited by 8 | Viewed by 989
Abstract
This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the [...] Read more.
This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with n=2, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
13 pages, 270 KiB  
Article
Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms
by Yanlin Li, Meraj Ali Khan, MD Aquib, Ibrahim Al-Dayel and Maged Zakaria Youssef
Axioms 2024, 13(3), 183; https://doi.org/10.3390/axioms13030183 - 11 Mar 2024
Cited by 10 | Viewed by 1650
Abstract
In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic [...] Read more.
In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)
34 pages, 558 KiB  
Article
D-Branes in Para-Hermitian Geometries
by Vincenzo Emilio Marotta and Richard J. Szabo
Universe 2022, 8(4), 200; https://doi.org/10.3390/universe8040200 - 23 Mar 2022
Cited by 6 | Viewed by 1719
Abstract
We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by [...] Read more.
We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds. Full article
(This article belongs to the Special Issue Dualities and Geometry)
12 pages, 235 KiB  
Article
Pseudo-Isotropic Centro-Affine Lorentzian Surfaces
by Olivier Birembaux
Mathematics 2020, 8(8), 1284; https://doi.org/10.3390/math8081284 - 4 Aug 2020
Cited by 5 | Viewed by 1715
Abstract
In this paper, we study centro-affine Lorentzian surfaces M2 in 3 which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that M2 is pseudo-isotropic if and only if the Tchebychev form T=0. In that case, [...] Read more.
In this paper, we study centro-affine Lorentzian surfaces M2 in 3 which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that M2 is pseudo-isotropic if and only if the Tchebychev form T=0. In that case, M2 is a an equi-affine sphere. Next, we will get a complete classification of centro-affine Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic. Full article
(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)
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