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Mathematics 2017, 5(4), 51; https://doi.org/10.3390/math5040051

Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
Received: 6 September 2017 / Revised: 6 September 2017 / Accepted: 10 October 2017 / Published: 16 October 2017
(This article belongs to the Special Issue Differential Geometry)
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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 purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons. View Full-Text
Keywords: Euclidean submanifold; position vector field; concurrent vector field; concircular vector field; rectifying submanifold; T-submanifolds; constant ratio submanifolds; Ricci soliton Euclidean submanifold; position vector field; concurrent vector field; concircular vector field; rectifying submanifold; T-submanifolds; constant ratio submanifolds; Ricci soliton
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Chen, B.-Y. Euclidean Submanifolds via Tangential Components of Their Position Vector Fields. Mathematics 2017, 5, 51.

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