On Jacobi-Type Vector Fields on Riemannian Manifolds

Abstract

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

A Note on Geodesic Vector Fields

“…[1][2][3]), Jacobi-type vector fields (cf. [4,5]), concircular vector fields (cf. [6][7][8][9]), torse forming vector fields (cf.…”

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

“…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. On the other hand, Matsuyama [24] derived a characterization stating that if the complete totally real submanifold n for the complex projective space CP n with bounded Ricci curvature admits a function ψ satisfying (3), for λ 1 ≤ n, then n is isometric to the hyperbolic space component that is connected if (∇ψ) x = 0 or if it is isometric to the warped product of a complete Riemannian manifold and the Euclidean line if ∇ψ is nonvanishing, where the warping function θ on R satisfies equation (2).…”

“…formula (10), along with the fact that ψ = λ 1 ψ, we haveB×{q}Hess(ψ)2 dV = -λ dV -B×{q} Ric(∇ψ, ∇ψ) dV (14). …”

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

“…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 . However, in [11], it has been proved that the complete totally real submanifold in CP n (complex projective space) with bounded Ricci curvature satisfying (3) is isometric to a special class of hyperbolic space.…”