\xi_\mu = g^{\lambda\nu}\left(
\tilde\nabla_\lambda g_{\nu\mu} - \frac12\tilde\nabla_\mu g_{\nu\lambda}\right), \qquad
\xi^\mu = g^{\lambda\nu}(\Gamma^\mu_{\lambda\nu} - \tilde\Gamma^\mu_{\lambda\nu}) = 
-\left(\frac g{\tilde g}\right)^{-1/2}\tilde\nabla_\nu\left(\left(\frac g{\tilde g}\right)^{1/2}g^{\mu\nu}\right),$$ where $\tilde\Gamma^\mu_{\lambda\nu}$ is the Levi-Civita connection for $\tilde g_{\mu\nu}$ and $\tilde\nabla_\mu$ is the associated covariant derivative.[^4] In the theory of harmonic maps between Riemannian manifolds, $\xi^\mu$ is referred to as the *tension field* of the identity map from $(M,g_{\mu\nu})$ to $(M,\tilde g_{\mu\nu})$, and the gauge-fixing condition $$\label{gfc}
\xi^\mu = 0$$ is equivalent to the requirement that the identity map be harmonic.

Ideally a gauge-fixing condition should intersect each gauge orbit exactly once: given a metric $\hat g_{\mu\nu}$, there should be a diffeomorphism (unique up to isometries) that maps it to a metric $g_{\mu\nu}$ satisfying . Such a diffeomorphism can be composed with the above identity map from $(M,g_{\mu\nu})$ to $(M,\tilde g_{\mu\nu})$ to obtain a harmonic map from $(M,\hat g_{\mu\nu})$ to $(M,\tilde g_{\mu\nu})$. In physicists’ language, this harmonic map is a solution to the classical equation of motion for the sigma model from $(M,\hat g_{\mu\nu})$ into $(M,\tilde g_{\mu\nu})$: $$\label{sigmamodel}
S[y] = \int_M \hat g^{1/2}\hat g^{\mu\nu}\tilde g_{\alpha\beta}(y)\partial_\mu y^\alpha\partial_\nu y^\beta\,,$$ where $y^\alpha$ are the coordinates on the target space $(M,\tilde g_{\mu\nu})$. At the local level, it is easy to see the existence and 