A characterization of an $m$-sphere ${\mathbf{S}}^m(a)$ is obtained using a non-trivial torse-forming vector field $\zeta$ on an $m$-dimensional Riemannian manifold.

The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.

We found conditions on an $n$-dimensional Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ to be trivial. First, we showed that under an appropriate upper bound on the squared length of the covariant derivative of the potential field $\mathbf{u}$, the Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ reduces to a trivial soliton. We also showed that appropriate upper and lower bounds on the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ of a compact Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ with potential field $\mathbf{u}$ geodesic vector field makes it a trivial soliton. We showed that if the Ricci operator $S$ of the Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ is invariant under the potential field $\mathbf{u}$, then $\left(M,g,\mathbf{u},\lambda \right)$ is trivial and the converse is also true. Finally, it was shown that if the potential field $\mathbf{u}$ of a connected Ricci soliton $\left(M,g,\mathbf{u},\lambda \right)$ is a concurrent vector field, then the Ricci soliton is shrinking.

We recall classical themes such as "on hearing the shape of a drum" or "can one hear the shape of a drum?", and the discovery of Milnor, who constructed two flat tori which are isospectral but not isometric. In this article, we consider the question of finding conditions under which an $n$-dimensional closed Riemannian manifold $\left(M^n,g\right)$ having a non-zero eigenvalue $nc$ for a positive constant $c$ (that is, has same non-zero eigenvalue as first non-zero eigenvalue of the sphere $S^n(c)$), is isometric to $S^n(c)$. We address this issue in two situations. First, we consider the compact $\left(M^n,g\right)$ as the hypersurface of the Euclidean space $\left(R^{n+1},⟨,⟩\right)$ with isometric immersion $f:\left(M^n,g\right)\to$ $\left(R^{n+1},⟨,⟩\right)$ and a constant unit vector $\overrightarrow{a}$ such that the function $\rho =⟨f,\overrightarrow{a}⟩$ satisfying $\mathrm{\Delta }\rho =-nc\rho$ for a positive constant $c$ is isometric to $S^n(c)$ if and only if $\left(M^n,g\right)$ is isometric to $S^n(c)$ provided the integral of Ricci curvature $Ric\left(\mathrm{\nabla }\rho ,\mathrm{\nabla }\rho \right)$ has an appropriate lower bound. In the second situation, we consider that the compact $\left(M^n,g\right)$ admits a non-trivial concircular vector field $\xi$ with potential function $\sigma$ satisfying $\mathrm{\Delta }\sigma =-nc\sigma$ for a positive constant $c$ and a specific function $f$ related to $\xi$ (called circular function) is constant along the integral curves of $\xi$ if and only if $\left(M^n,g\right)$ is isometric to $S^n(c)$.

In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field $\omega$ on a compact and connected hypersurface $N$ of the Euclidean space $R^{m+1}$ with a mean curvature $\alpha$ constant along the integral curves of $\omega$ and a shape operator $T$ satisfying $T(\omega )=\alpha \omega$ implies that $\alpha$ is a constant and $N$ is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field $\mathbf{v}$ on a compact and connected hypersurface $N$ of a Euclidean space $R^{m+1}$ gives a nonzero function $\sigma =g\left(T\mathbf{v},\mathbf{v}\right)$ with shape operator $T$, and the integral of the function $m\alpha \sigma Ric\left(\mathbf{v},\mathbf{v}\right)$ has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface $N$ with support $\rho$ and basic vector field $\mathbf{u}$, the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.