In this thesis, we summarize some of the characterization theorems of the sphere, the content of Engman’s paper [6]. We investigate the method used by Cheng [4] and we can find out that there is another method to characterize the sphere using eigenfunctions. We discover that if f is an eigenfunction of the Laplacian, with f satisfying the relation f² + c₁g(∇f,∇f) = c₂ for some constant c₁ and c₂, then the manifold must be isometric to the standard sphere or the flat torus, depending on its topology. ^{n 2} _S ⁿ...[ Read more ]

In this thesis, we summarize some of the characterization theorems of the sphere, the content of Engman’s paper [6]. We investigate the method used by Cheng [4] and we can find out that there is another method to characterize the sphere using eigenfunctions. We discover that if f is an eigenfunction of the Laplacian, with f satisfying the relation f² + c₁g(∇f,∇f) = c₂ for some constant c₁ and c₂, then the manifold must be isometric to the standard sphere or the flat torus, depending on its topology.

Next we investigate Engman’s method to surface of revolution and try to extend it into higher dimensions. We try to extend the metric on Sⁿ for n ≥ 3, with metric g = ds ⊗ ds + [r(s)]²can_Sn-1 on Sⁿ {n, s}. If the method of classical separation of variables is valid, then we can find out that the multiplicities of sphere must be the highest, compared with other kinds of “spherical revolutions”.