On a differential equation characterizing Euclidean spheres.
Obata showed that a necessary and sufficient condition for a connected, complete Riemannian manifold \((M,g)\) to be isometric with the Euclidean sphere of radius \(1/\sqrt{\lambda},\) \( \lambda>0 \) is the existence of a nonconstant function \(f\) on \(M\) satisfying the differential equation \(H_{f}+\lambda f g =0,\) where \(H_{f}\) is the Hessian of f on \((M,g).\) While, the differential equation \(1/\sqrt{\lambda},\) \(\lambda>0 \) on a connected, complete, Riemannian manifold \((M,g)\) has a nontrivial solution iff its domain \((M,g)\) is the Euclidean sphere of radius \(1/\sqrt{\lambda}.\) In this paper, the authors show that a necessary and sufficient condition for a connected, complete \(n(\geq2)\)-dimensional Riemannian manifold to be isometric with the Euclidean sphere of radius \(1/\sqrt{\lambda},\) \(\lambda>0,\) is the existence of a nonzero vector field \(Z\) on \((M,g)\) satisfying the differential equation \((\nabla \nabla Z) (X,Y) + \lambda g (Z,X) (Y)=0\) for all \(X,Y \in \Gamma(TM),\) where \(\nabla\) is the covariant derivative on \(M\). They also analyze the case \(\lambda \leq 0\) for the differential equation \((\nabla \nabla Z) (X,Y) + \lambda g (Z,X) (Y)=0\) for all \(X,Y \in \Gamma(TM),\) on a connected complete Riemannian manifold \((M,g).\)
scientific article; zbMATH DE number 2005880