Abstract
In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold ( M , g ) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n ( c ) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold.
Highlights
One of the important topics in differential geometry of a Riemannian manifold (M, g) is the study of the influence of special vector fields on its geometry as well as topology
It is well known that their existence has considerable impact on the geometry of the Riemannian manifold and these vector fields are used in finding characterizations of spheres as well as Euclidean spaces
In [11], Fialkow initiated the study of concircular vector fields on a Riemannian manifold
Summary
One of the important topics in differential geometry of a Riemannian manifold (M, g) is the study of the influence of special vector fields on its geometry as well as topology. These special vector fields are geodesic vector fields, Killing vector fields, concircular vector fields, Jacobi-type vector fields, and conformal vector fields on a Riemannian manifold. A concircular vector field u is said to be a non-trivial concircular vector field if the potential function f , 0. We answer this question by showing that to each non-trivial concircular vector u with potential function. In the last section, we observe that the connecting function ρ influences topology of the Riemannian manifold (cf Theorems 7 and 8)
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