Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

Chen, Bang‐Yen

doi:10.3390/math5040051

Cited by 10 publications

(6 citation statements)

References 38 publications

(47 reference statements)

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“…It was proved in [10] that the gradient of a non-constant smooth function σ on a Riemannian manifold (M, g) is a torse-forming vector field with…”

Section: Almost η-Ricci Solitonsmentioning

confidence: 99%

“…Let M be an n-dimensional hypersurface isometrically immersed into an (n + 1)-dimensional Riemannian manifold M , g . If V is a torse-forming vector field on M and η is the g-dual of V T , then i) (M, g) is an almost η-Ricci soliton with potential vector field V T if and only if there exist two smooth functions λ and µ on M such that the Ricci tensor field of M satisfies (10) Ric…”

Section: Almost η-Yamabe Solitonsmentioning

confidence: 99%

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Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

Blaga¹,

Özgür²

2020

Preprint

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We provide properties of almost η-Ricci and almost η-Yamabe solitons on submanifolds isometrically immersed into a Riemannian manifold M , g whose potential vector field is the tangential component of a torse-forming vector field on M , treating also the case of a minimal or pseudo quasi-umbilical hypersurface. Moreover, we give necessary and sufficient conditions for an orientable hypersurface of the unit sphere to be an almost η-Ricci or an almost η-Yamabe soliton in terms of the second fundamental tensor field. PreliminariesLet g be a (pseudo)-Riemannian metric on an n-dimensional smooth manifold M, Ric its Ricci curvature tensor field, scal its scalar curvature, V a vector field on M, £ V the Lie derivative in the direction of V and η a 1-form on M.If there exist two smooth functions λ and µ on M such thatthen (M, g) is said to be an almost η-Ricci soliton [4] and we denote it by (V, λ, µ).

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“…It was proved in [10] that the gradient of a non-constant smooth function σ on a Riemannian manifold (M, g) is a torse-forming vector field with…”

Section: Almost η-Ricci Solitonsmentioning

confidence: 99%

Section: Almost η-Yamabe Solitonsmentioning

confidence: 99%

Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

Blaga¹,

Özgür²

2020

Preprint

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“…In earlier articles, we have investigated Euclidean submanifolds whose canonical vector fields are concurrent [6,8], concircular [14], torse-forming [13], conformal [12], or incompressible [11]. (See also recent surveys [9,10] for several topics on position vector fields on Euclidean submanifolds. )…”

Section: Incompressible Vector Fieldsmentioning

confidence: 99%

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

Chen

2018

Tamkang J. Math.

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Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.

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“…In earlier articles, we have investigated Euclidean submanifolds whose canonical vector fields are concurrent [5,6], concircular [12], torseforming [11], or conformal [10]. (See [8,9] for recent surveys on several topics associated with position vector fields on Euclidean submanifolds. )…”

Section: Introductionmentioning

confidence: 99%

Euclidean submanifolds with incompressible canonical vector field

Chen

2018

Preprint

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For a submanifold M in a Euclidean space E m , the tangential component x T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M . In this article, first we prove that the canonical vector field of every Euclidean submanifold is always conservative. Then we initiate the study of Euclidean submanifolds with incompressible canonical vector fields. In particular, we obtain the necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible. Further, we provide examples of Euclidean submanifolds with incompressible canonical vector field. Moreover, we classify planar curves, surfaces of revolution and hypercylinders with incompressible canonical vector fields.

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Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

Cited by 10 publications

References 38 publications

Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

Almost $η$-Ricci and almost $η$-Yamabe solitons with torse-forming potential vector field

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

Euclidean submanifolds with incompressible canonical vector field

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