Abstract
In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.
Highlights
Differential equations on complete and connected Riemannian manifolds were used by Obata, who observed that a necessary and sufficient condition for an m-dimensional complete and connected Riemannian manifold ( M, g) to be isometric to m-sphere Sm (c) of constant curvature c is that it admits a nontrivial solution of the differential equation
In [5], it has been observed that a necessary and sufficient condition for an m-dimensional complete and connected Riemannian manifold ( M, g) to be isometric to the Euclidean space Em is that it admits a nontrivial solution of the differential equation
Differential equations satisfied by certain vector fields such as conformal vector fields, Killing vector fields, Jacobi-type vector fields, torse-forming vector fields, geodesic vector fields are useful in studying geometry of a Riemannian manifolds
Summary
Differential equations on complete and connected Riemannian manifolds were used by Obata (cf. [1,2]), who observed that a necessary and sufficient condition for an m-dimensional complete and connected Riemannian manifold ( M, g) to be isometric to m-sphere Sm (c) of constant curvature c is that it admits a nontrivial solution of the differential equation. In [5], it has been observed that a necessary and sufficient condition for an m-dimensional complete and connected Riemannian manifold ( M, g) to be isometric to the Euclidean space Em is that it admits a nontrivial solution of the differential equation. Recall that a smooth vector field ξ on a Riemannian manifold ( M, g ) is called a concircular vector field if it satisfies the differential equation. A smooth vector field ξ on a Riemannian manifold ( M, g) is called torse-forming vector field if it satisfies the differential equation. We find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field In the last section, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field
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