@article{Deshmukh2024, 
author = {Sharief Deshmukh and Amira Ishan and Olga Belova},
title = {On eigenfunctions corresponding to first non-zero eigenvalue of the sphere  Sn(c) on a Riemannian manifold},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {12},
pages = {34272-34288},
keywords = {first non-zero eigenvalue, eigenfunction, Laplace operator, Ricci curvature, isometric to sphere},
url = {https://www.sciopen.com/article/10.3934/math.20241633},
doi = {10.3934/math.20241633},
abstract = {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  (Mn,g) 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  Sn(c)), is isometric to  Sn(c). We address this issue in two situations. First, we consider the compact  (Mn,g) as the hypersurface of the Euclidean space  (Rn+1,⟨,⟩) with isometric immersion  f:(Mn,g)→  (Rn+1,⟨,⟩) and a constant unit vector  a→ such that the function  ρ=⟨f,a→⟩ satisfying  Δρ=−ncρ for a positive constant  c is isometric to  Sn(c) if and only if  (Mn,g) is isometric to  Sn(c) provided the integral of Ricci curvature  Ric(∇ρ,∇ρ) has an appropriate lower bound. In the second situation, we consider that the compact  (Mn,g) admits a non-trivial concircular vector field  ξ with potential function  σ satisfying  Δσ=−ncσ for a positive constant  c and a specific function  f related to  ξ (called circular function) is constant along the integral curves of  ξ if and only if  (Mn,g) is isometric to  Sn(c).}
}