On Killing Vector Fields on Riemannian Manifolds

Abstract

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

Generalized Ricci solitons and Einstein metrics on weak $K$-contact manifolds

“…For a weak contact metric manifold, the tensors N (2) and N (4) vanish and the integral curves of ξ are geodesics; moreover, N (3) vanishes if and only if ξ is a Killing vector field, see [8].…”

“…metric contact manifolds whose characteristic vector field generates a 1-parameter group of isometries) have been studied by several geometers, e.g., [1,11], and it is seen that the K-contact structure is intermediate between the contact and Sasakian structures. The characteristic vector field ξ of the K-contact structure is a unit Killing vector field, and the influence of constant length Killing vector fields on the geometry of Riemannian manifolds has been studied by several authors from different points of view, e.g., [3,4,6]. An interesting result related to the above question is that a K-contact manifold equipped with generalised Ricci soliton structure has an Einstein metric, e.g., [5].…”

Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons

“…where X i ∈ X M and • denotes the operator of omission, defines a (k + 1)-form dω called the exterior differential of ω. Note that ( 7) and ( 11) correspond to (12) with k = 1 and k = 2.…”

“…an f -contact structure, whose characteristic vector fields are unit Killing vector fields, see [18], can be regarded as intermediate between a metric f -structure and S-structure (the Sasaki structure when s = 1). The influence of constant length Killing vector fields on the geometry of Riemannian manifolds has been studied by several authors from different points of view, e.g., [2,12]. The curvature of f -contact and f -K-contact manifolds was studied in [6,29,30].…”

Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q

“…As it is well-known, the description of Hamiltonian mechanics is developed on symplectic manifolds. Contact geometry has been applied to give a Hamiltonian-type description of mechanical systems with dissipation [3], field theories, and gravitation in an odd number of dimensions, Sasaki-Einstein geometries [4,5]. An analogous theory to complete integrability in symplectic geometry was constructed in contact geometry [6,7].…”