# Question involving curvature tensor

1. ### Kalimaa23

307
Greetings,

I'm working out some of the mathematical relations between Yang-Mill theory and GR. I'm having difficulty working out a somewhat trivial thing, I was wondering if anyone here could help me.

To keep things concrete, I'll stick to the case D=4, but I'd like to be able to generalise to higher dimensional cases afterwards.

Consider the Riemann curvature tensor. It has 20 independent components. The vacuüm Einstein equations amount to setting the Ricci tensor to zero, which imposes 10 constraints on the Riemann tensor. The additional 10 components are then described the the Weyl-tensor.

Now imaging that we do not impose the Einstein equations. How many constraints would the equation

$$D^{\mu} R_{\mu\nu\sigma\tau}=0$$ (1)

impose, together with the second bianchi identity

$$D_{\left[\rho\right.}R_{\left.\mu\nu\right]\sigma\tau}=&0$$ (B2)
?

The reason I'm asking is because I've proven that (1)+(B2) implies Einstein equations. I'm wondering if the reverse if true, and if not, what additional constraints must be applied.