Help finding metric from $\frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})=K^k g_{ij}$

Question

I need to find a metric in $\Bbb R^4$ from

$$ \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})=K^k g_{ij}$$

Where $K^k$ are arbitrary real constants and we are using einstein notation. It basically asks when a levi-civita connection has the coefficients proportional to the metric.

I started seeing what happens when all indices are equal and I got this(I hope it is right):

$$\Gamma_{ii}^i = g^{il}\partial_i g_{il} - \frac12\partial^i g_{ii}=K^i g_{ii}$$

I don't know what more I can do to find at least one valid metric. Thanks.

Answer 1

You should remember the following formulas $$ g^{il}g_{im}=\delta^l_m \qquad g^{ij}g_{ij}=D $$ where $D$ is the dimension of the manifold. Then, $$ K^kg_{ij}g^{ij}=\frac{1}{2}g^{ij}g^{kl}\left(\partial_ig_{jl}+\partial_jg_{il}-\partial_lg_{ij}\right) $$ that yields $$ K^k=D^{-1}\partial_ig^{ik}, $$ where use has been made of the equation $\partial_l(g^{ij}g_{ij})=0$.