A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

Abstract

In this paper, we show that given a nontrivial 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 , which we call the connecting function of the concircular vector field u. Then 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 topology of the Riemannian manifold.2000 Mathematics Subject Classification. Primary 53C20, 53C21.

The base of warped product submanifolds of Sasakian space forms characterized by differential equations

“…Such complete space classifications are extremely attractive and have been studied by several mathematicians (see, e.g., [3,4,6,7,15,16,[18][19][20]). For example, by using (1), Al, Dayel, Deshmukh, and Belova [1] showed that a connected and complete Riemannian manifold ( n , g) is isometric to R n if and only if the nontrivial concircular vector field u along the function ψ satisfies R(∇ψ, ∇ψ) = 0 or u = 0. In [13], Chen and Deshmukh proved that a complete Riemannian manifold admits a concurrent vector field if and only if it is isometric to a Euclidean space by (1).…”

“…For example, by using (1), Al, Dayel, Deshmukh, and Belova [1] showed that a connected and complete Riemannian manifold ( n , g) is isometric to R n if and only if the nontrivial concircular vector field u along the function ψ satisfies R(∇ψ, ∇ψ) = 0 or u = 0. In [13], Chen and Deshmukh proved that a complete Riemannian manifold admits a concurrent vector field if and only if it is isometric to a Euclidean space by (1). Similarly, in [14], it has been shown that ( n , g) is isometric to a Euclidean space if and only if ( n , g) permits a nontrivial gradient conformal vector field, that is, a Jacobi-type vector field.…”

Characterization of Skew CR-Warped Product Submanifolds in Complex Space Forms via Differential Equations

“…e categorization of differential equations on Riemannian manifold has become a fascinating topic of research and has been investigated by numerous researchers, for instance, [6][7][8][9].…”

“…Recently, Al-Dayel et al [6] studied the impact of differential equation (2) on Riemannian manifold (L n , g) by taking the concircular vector field and proved that, under certain conditions, the Riemannian manifold (L n , g) is isometric to Euclidean manifold R n . Similarly, by taking gradient conformal vector field, Chen et al [10] identified that Riemannian manifold (N n , g) is isometric to the Euclidean space R n .…”

Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms

“…The categorization of differential equations on Riemannian manifolds turns into an attractive research subject that has been explored by various researchers, for example, [7][8][9][10][11].…”