Resultados de descomposición asociados á ecuación de Möbius

In analysis, mostly the existence of a nontrivial solution to a differential equation on a certain domain is argued. But in geometry, one can also argue the existence of a manifold structure for a differential equation to possess a nontrivial solution. This may be considered as an analytic characterization (or representation) of a manifold structure by a differential equation if this manifold structure serves as a unique domain structure for this differential equation to possess a nontrivial solution in a certain class of manifolds. An example to the previous program is Obata’s Theorem, which completely characterizes Euclidean spheres among compact Einstein manifolds by the existence of a solution of the differential equation Δφ = −nkφ, where φ is a real-valued function on the manifold M, n = dimM and k denotes the reduced scalar curvature