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 \(\sigma \) and \(\rho \) 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 \( {\mathbf {S}}^{n}(c)\). 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 \( {\mathbf {S}}^{n}(c)\). Similarly, we find two characterizations of Euclidean spaces \({\mathbf {E}}^{n}\). 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 \({\mathbf {E}}^{n}\).

在本文中，我们展示了非平凡的圆周向量场在表征球体以及欧几里得空间方面起着重要作用。给定连通黎曼流形 ( M ,  g )上的非平凡圆周向量场\(\xi \)，两个称为势函数和连通函数的光滑函数\(\sigma \)和\(\rho \)自然关联到\(\xi \)。我们在n维紧凑黎曼流形上使用非平凡的圆周向量场来找到球体\( {\mathbf {S}}^{n}(c)\) 的四种不同特征。特别地，我们证明了一个有趣的结果，即n一维紧致黎曼流形 ( M ,  g ) 允许一个非平凡的圆周向量场\(\xi \)使得 Ricci 算子在\(\xi \)流下是不变的，当且仅当， ( M ,  g ) 等距于球体\( {\mathbf {S}}^{n}(c)\)。类似地，我们发现欧几里得空间\({\mathbf {E}}^{n}\) 的两个特征。特别地，我们证明了一个n维完备且连通的黎曼流形 ( M ,  g ) 承认一个非平凡的圆周向量场\(\xi \)消除 Ricci 算子，当且仅当 ( M ,  g ) 与欧几里得空间等距\({\mathbf {E}}^{n}\)。