Exercise 3.01 Consequences of metric compatibility

Christoffel Symbol

At last I am on to chapter 3 on curvature. It has taken me three weeks to do the first four pages and exercise 3.01 was done on the way. I learned or relearned many things and had two interesting discussions on Physics forums (here and here). The question asked us to prove that the covariant derivative of the inverse metric and and that the covariant derivative of the Levi-Civita tensor both vanished if the covariant derivative of the metric vanishes and the Christoffel Symbol is symmetric in its lower indices. These conditions are known as metric compatibility and torsion freedom(?) respectively. One would say 'a connection on a manifold is torsion free and metric compatible'.



In maths that is

Question

Verify the consequences of metric compatibility: If
\begin{align}
{\mathrm{\nabla }}_{\sigma}g_{\mu \nu }=0 & \phantom {10000}(1) \\
\end{align}then (a)\begin{align}
{\mathrm{\nabla }}_{\sigma}g^{\mu \nu }=0 & \phantom {10000}(2) \\
\end{align}and (b)\begin{align}
{\mathrm{\nabla }}_{\lambda}{\varepsilon }_{\mu \nu \sigma \rho }=0 & \phantom {10000}(3) \\
\end{align} I am not sure if we are assuming ##{\mathrm{\Gamma }}^{\tau }_{\lambda \mu }={\mathrm{\Gamma }}^{\tau }_{ \mu \lambda }## or not.

Answer

Part (a) was quite simple but I struggled with the part (b) until 23 March and had to give up. Along the way I had lots of practice at index manipulation, I reacquainted myself with Cramer's rule for solving simultaneous equations, proved (b) on the surface of a sphere, found the 'dynamite' version of Carroll's streamlined matrix determinant equation (2.66) and added some equation shortcut keys to my keyboard. The time was not wasted.

We make frequent use here of  the fact that ##g_{\mu \nu }g^{\mu \rho }\mathrm{=}{\delta}^{\rho }_{\nu }## and the indexing effect of the Kronecker delta: ##{\delta}^{\lambda }_{\beta }\mathrm{\Gamma }^{\mu }_{\sigma \lambda }={\mathrm{\Gamma }}^{\mu }_{\sigma \beta }## because we are summing over ##\lambda ## and the only non-zero term is when ##\beta =\lambda ##. In this case ##\mathrm{\Gamma }## can be replaced by any symbol or tensor of any rank.

Here is the full effort Ex 3.01 Consequences of metric compatibility.pdf (7 pages of which 4 might be worth looking at).

The comment left by JSBach1801 solved this problem very easily as I fully realised in Jan 2021. Thanks! That answer is right at the end of the pdf.