Let (M,g) be a riemannian surface with a family of metric connections ∇. As the connection is metric, the curvature R(X,Y) is antisymmetric so that we have proper values ωi and vectors ei:
R(X,Y)ek=iωk(X,Y)ek
with ωk∈Λ2(TM). The proper values can be viewed as antisymmetric endomorphisms by the metric Ωk∈End(TM). A flow of curvature can be defined by the formula:
∂g∂t(X,Y)=−∑kg(ΩkX,ΩkY)
Can we have solutions of the flow for small times ?