Characterizing specific Riemannian manifolds by differential equations
Let \((M,g)\) be a connected complete Riemannian manifold of dimension \(m\geq2\). Obata showed the standard sphere of constant sectional curvature \(k>0\) is characterized as being the only manifold where there exists a non-constant function satisfying the equation \(H_f+kfg=0\). The authors present an elegant survey of the field discussing characterizations of spheres, flat spaces, hyperbolic spaces, complex projective spaces, etc. as domains where certain natural equations on certain tensors have solutions analogous to that given above by Obata. The paper contains a lovely bibliography of the field.
scientific article; zbMATH DE number 1924876