2017
DOI: 10.3390/math5040051
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Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

Abstract: Abstract:The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x(t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The p… Show more

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Cited by 12 publications
(8 citation 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%
See 1 more Smart Citation
“…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%
“…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%
“…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%