A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

Chen, Bang‐Yen

doi:10.5556/j.tkjm.49.2018.2804

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…For further results on submanifolds with incompressible canonical vector fields, see [83]. A vector field 𝑣𝑣𝑣𝑣 on a Riemannian manifold 𝑀𝑀𝑀𝑀 is called a conformal vector field if it satisfies:…”

Section: Differential Geometry Of Canonical Vector Fieldsmentioning

confidence: 99%

Differential Geometry of Position Vector Fields

Chen

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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Differential geometry studies the geometry of curves, surfaces, and higher dimensional smooth manifolds. It uses the ideas and techniques of differential and integral calculus, linear and multilinear algebras, topology, and differential equations. This subject in mathematics is closely related to differential topology, which concerns itself with properties of smooth manifolds. Differential geometry also closely relates to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.Curvature is an important notion in mathematics, which has been investigated extensively in differential geometry. There are two types of curvatures: namely, "intrinsic" and "extrinsic"."Intrinsic curvature" describes the curvature at a point on a surface or a smooth manifold and is independent of how the surface or manifold is embedded in space. Borrow a term from biology, intrinsic invariants of a manifold are the DNA of the manifold. The Gauss curvature of a surface is the most commonly studied intrinsic measure of curvature. In higher dimensions, curvature is too complicated to be described by a single number. In this case, tensors are used to describe the curvature as pioneered by B. Riemann in his famous 1854 inaugural lecture at Gottingen: "Über die Hypothesen welche der Geometrie zu Grunde liegen."In Einstein's theory of general relativity, intrinsic curvature is key to understanding the shape of the universe."Extrinsic curvature" of a manifold depends on how it is embedded within a space. Examples of extrinsic measures of curvature include geodesic curvature, principal curvature, and mean curvature. The most important extrinsic invariant for a submanifolds in an ambient Riemannian manifold is the mean curvature vector, which is known as the tension field in physics.Differential geometry studies the geometry of curves, surfaces and higher dimensional smooth manifolds. For submanifolds in Euclidean spaces, the position vector is the most natural geometric object. Position vectors find applications throughout mathematics, engineering and natural sciences. The purpose of this survey article is to present six research topics in differential geometry in which the position vector plays a very important role. In addition to this, we explain the link between position vectors with mechanics, dynamics, and D'Arcy Thompson's law of natural growth in biology.

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Section: Differential Geometry Of Canonical Vector Fieldsmentioning

confidence: 99%

Differential Geometry of Position Vector Fields

Chen

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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“…Incompressible vector fields are very important in magnetohydrodynamics and they are often used in modern technology, especially in electronic engineering and electrodynamics (cf. [1][2][3][4][5][6]).…”

Section: Introductionmentioning

confidence: 99%

A Note on Incompressible Vector Fields

Turki

2023

Symmetry

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In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

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“…Chen and his co-author studied Euclidean submanifold whose canonical vector field are concurrent [4], concircular [11], conformal [10], torse-forming [9] and also in ( [3], [5], [6]). Ricci soliton and Yamabe soliton whose canonical vector field are concurrent and conformal studied in ([2], [7], [8]).…”

Section: Introductionmentioning

confidence: 99%