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Keywords = Obata connection

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17 pages, 294 KB  
Article
The Obata-Type Connection on a 3-Parameter Generalized Quaternion Structure
by Bogdan Balcerzak
Symmetry 2025, 17(9), 1501; https://doi.org/10.3390/sym17091501 - 10 Sep 2025
Viewed by 446
Abstract
This paper aims to define the Obata-type connection on a 4n-dimensional smooth manifold equipped with a 3-parameter generalized quaternion structure. The paper demonstrates some general algebraic properties of this connection. The integrability of endomorphisms of the tangent bundle is interpreted as [...] Read more.
This paper aims to define the Obata-type connection on a 4n-dimensional smooth manifold equipped with a 3-parameter generalized quaternion structure. The paper demonstrates some general algebraic properties of this connection. The integrability of endomorphisms of the tangent bundle is interpreted as homomorphisms of certain skew-symmetric algebroids. The results confirm the uniqueness of a torsion-free connection compatible with generalized quaternion structures, assuming their integrability for non-zero parameters. Full article
(This article belongs to the Section Mathematics)
11 pages, 263 KB  
Article
Application of Differential Equations on the Ricci Curvature of Contact CR-Warped Product Submanifolds of S2n+1(1) with Semi-Symmetric Metric Connection
by Meraj Ali Khan, Amira A. Ishan, Ibrahim Al-Dayel and Khalid Masood
Symmetry 2024, 16(11), 1463; https://doi.org/10.3390/sym16111463 - 4 Nov 2024
Viewed by 970
Abstract
In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric [...] Read more.
In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric sphere. Additionally, we investigate the impact of specific differential equations on these submanifolds and demonstrate that, when certain geometric conditions are met, the base submanifold can be characterized as a special type of warped product. Full article
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