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Spheres and Euclidean Spaces Via Concircular Vector Fields
(Springer Basel Ag, 2021) Deshmukh, Sharief; Ilarslan, Kazim; Alsodais, Hana; De, Uday Chand
In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field xi on a connected Riemannian manifold (M, g), two smooth functions s and. called potential function and connecting function are naturally associated to xi. We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres S-n (c). In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field xi such that the Ricci operator is invariant under the flow of xi, if and only if, (M, g) is isometric to a sphere Sn (c). Similarly, we find two characterizations of Euclidean spaces E-n. In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field xi that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space E-n.