The determinant of a metric makes perfectly good sense, but it is not a function, rather a $2$-density. Formally, this means that it transforms as a section of the bundle associated with the frame bundle and a particular nontrivial character of the general linear group.

Let $\phi:V \to W$ be a linear map between $n$-dimensional vector spaces $V$ and $W$ (for simplicity, over a field of characteristic $0$). The induced map $\wedge^{n}\phi:\wedge^{n}V \to \wedge^{n}W$ can be identified with multiplication by a scalar when a generator is chosen for each of $\wedge^{n}V$ and $\wedge^{n}W$. When $V = W$, and the same generator is used on either side, the resulting scalar does not depend on the choice, and it is reasonable to call this scalar $\det \phi$ because it agrees with the usual determinant of an endomorphism.

A symmetric bilinear form $g$ on $V$ is identified with a linear map $V \to V^{\ast}$ to the dual vector space. The determinant $\det g$ is defined as in the preceding paragraph, as the induced map $\wedge^{n}V \to \wedge^{n}V^{\ast}$.

Consider the standard action of $GL(n) = GL(V)$ on $V$ and the induced actions on $V^{\ast}$ and tensor powers of $V$ and $V^{\ast}$. So the action of $\gamma \in GL(n)$ on $g$ is given by $(\gamma \cdot g)(u, v) = g(\gamma^{-1}\cdot u, \gamma^{-1}\cdot v)$ for $u,v \in V$. It follows straightforwardly that $\det (\gamma \cdot g) = (\det \gamma)^{-2}\det g$ for $\gamma \in GL(n)$.

Suppose the base field is $\mathbb{R}$. Thus $\det:S^{2}V^{\ast} \to \mathbb{R}$ is a $GL(n)$-equivariant map for the standard action of $GL(n)$ on $S^{2}V^{\ast}$ and the $1$-dimensional representation $\chi:GL(n) \to GL(1)$ given by $\chi(\gamma) = (\det \gamma)^{-2}$.

Now let $M$ be a smooth $n$-dimensional manifold with frame bundle $F\to M$. With each $GL(n)$ module $(\rho, W)$ there is associated a bundle of weighted tensors $F \times_{\rho}W$ whose fibers are linearly isomorphic to $W$. Applying this construction to the representations of the preceding paragraph one obtains a map $\det$ associating with a section of $S^{2}T^{\ast}M$ a section of the line bundle associated with the representation $\chi$, which can be interpreted as the tensor square of the top exterior power $\wedge^{n}T^{\ast}M$ (such a section is often called a $2$-density).

Let $g$ be a Riemannian metric on $M$ with Levi-Civita connection $\nabla$. Let $h$ be a section of $S^{2}T^{\ast}M$. Picking a local frame $\{E_{1}, \dots, E_{n}\}$ in $TM$ determines a local trivialization of $\wedge^{n}T^{\ast}M$ so also of all its tensor powers. With respect to this trivialization, $\det h$ equals the determinant of the matrix $h(E_{i}, E_{j})$. The connection determined by $g$ determines a convariant derivative on any associated bundle of the frame bundle and the covariant derivative $\nabla \det h$ is a section of the same line bundle as is $\det h$. If $h$ is everywhere full rank and $h^{-1}$ is the section of $S^{2}TM$ inverse to $h$, then $\nabla \det h = (h^{-1}\nabla h)\det h$ (the notation requires interpretation; in abstract index notation $h^{-1}\nabla h$ means $h^{pq}\nabla_{i}h_{pq}$). In particular, $\nabla \det g = 0$ because $\nabla g = 0$.

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