Let (M,g) be a riemannian manifold with riemannian curavature R∈Λ2(TM)⊗End(TM), then I define the R2-curvature:
R2=R(ei,ej)R(ei,ej)
with an orthonormal basis (ei). R2 is symmetric and negativ.
We can then define the Einstein equations for R2-curvature:
R2=−Id
I define:
r2(X,Y)=g(R2(X),Y)
The r2-curvature flow is then:
∂g∂t=r2
Can we have solutions for the r2 flow?