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.