On eigenfunctions corresponding to first non-zero eigenvalue of the sphere $S^n(c)$ on a Riemannian manifold

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 $S^n(c)$.