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}, \langle, \rangle \right) $ with isometric immersion $ f:\left(M^{n}, g\right) \rightarrow $ $ \left(R^{n+1}, \langle, \rangle \right) $ and a constant unit vector $ \overrightarrow{a} $ such that the function $ \rho = \langle f, \overrightarrow{a}\rangle $ satisfying $ \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(\nabla \rho, \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 $ \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) $.