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.
Highlights
For an n-dimensional submanifold M in the Euclidean m-space Em, the most elementary and natural geometric object is the position vector field x of M
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: its location relative to a given coordinate system at some time variable t
We will discuss constant-ratio submanifolds, as well as Ricci solitons on Euclidean submanifolds with the potential fields of the Ricci solitons coming from the tangential components of the position vector fields
Summary
For an n-dimensional submanifold M in the Euclidean m-space Em , the most elementary and natural geometric object is the position vector field x of M. The position vector is a Euclidean vector. The first and the second derivatives of the position vector field with respect to time t give the velocity and acceleration of the particle. For a Euclidean submanifold M of a Euclidean m-space, there is a natural decomposition of the position vector field x given by:. We discuss Euclidean submanifolds M whose tangential components x T admit some special properties such as concurrent, concircular, torse-forming, etc. We will discuss constant-ratio submanifolds, as well as Ricci solitons on Euclidean submanifolds with the potential fields of the Ricci solitons coming from the tangential components of the position vector fields.
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