The meaning of "metric preserving" connection.

Question

Let $g$ be a metric on a smooth manifold $M$. In local coordinates the metric takes the form $$g=g_{ij}(dx^i\otimes dx^j+dx^j\otimes dx^i).$$ From here, a connection $\nabla$ on $M$ is said to be metric preserving if $\nabla g=0$. But what is the quantity $\nabla g$?

Answer 1

Suppose we take a curve $\gamma$ through a point $p\in M$. Take two vectors $v$, $w\in T_p M$. Let $V$ and $W$ be the parallel vector fields along $\gamma$ such that $V(p)=v$ and $W(p)=w$. So $\nabla_{\gamma'}V=0$ and $\nabla_{\gamma'}W=0$.

If $\nabla g = 0$, then $$ \frac{d}{dt}g(V,W) = g(\nabla_{\gamma'}V, W) + g(V,\nabla_{\gamma'}W) = 0. $$ So the inproduct $g(V,W)$ is constant along parallel transport if $\nabla$ is compatible with the metric, i.e. $\nabla g=0$. To phrase it a bit more informal: parallel transport preserves lengths and angles.

If $\nabla g\neq 0$, then $$ \frac{d}{dt}g(V,W) = (\nabla_{\gamma'} g)(V,W) + g(\nabla_{\gamma'}V, W) + g(V,\nabla_{\gamma'}W) = (\nabla_{\gamma'} g)(V,W). $$ So the quantity $(\nabla_{\gamma'} g)(V,W)$ gives the change of the inner product of $V$ and $W$ along the curve.

Answer 2

$\nabla$ is called a covariant derivative. It is a bilinear transformation $T_pM \times T_n^m(M) \ni (v,u) \mapsto \nabla_vu \in T_n^m(T_pM) $, that satisfies some specific properties, most notably the Leibniz rule $\nabla_v (fu) = (\nabla_v f) u + f (\nabla_v u)$. For every class of tensors it is defined as follows:

On functions $\nabla_v f$ is just usual directional derivative: $$\nabla_v f = i_v df = \langle df,v\rangle $$

On vector fields $\nabla_v u$ is usually assumed to be given and it used to define a connection by defining the parallel transport along a curve as such for which the covariant derivative along the curve vanishes. If you have the connection given in another way, you can reverse this definition and calculate the covariant derivative knowing the parallel transport function.

On 1-forms $\nabla_v$ is defined by the condition $$\forall X\in\mathfrak{X}(M):\nabla_v \langle \omega, X\rangle = \langle \nabla_v\omega, X\rangle + \langle \omega, \nabla_vX\rangle $$ that is $\nabla_v\omega$ is a 1-form such that for any vector field $X\in\mathfrak{X}(M)$: $$ \langle \nabla_v\omega, X\rangle = \nabla_v \langle \omega, X\rangle - \langle \omega, \nabla_vX\rangle$$

On more complicated tensors $\nabla_v$ is defined by the Leibniz rule: $$ \nabla_v (u \otimes w) = (\nabla_v u) \otimes w + u \otimes (\nabla_v w) $$

In particular, for any point $p$ and any vector $v\in T_pM$, $\nabla_v g \in T^0_2(T_pM)$ is a tensor such that for any vector fields $X,Y\in \mathfrak{X}(M)$ $$ \langle \nabla_vg, X\otimes Y\rangle = \nabla_v \langle g, X\otimes Y\rangle -\langle g, \nabla_vX\otimes Y\rangle - \langle g, X\otimes \nabla_vY\rangle$$

$\nabla g$ itself is a function $TM \supset T_pM \ni v \mapsto \nabla_v g \in T^0_2(T_pM)$, which can also bea seen as a tensor field $\nabla g \in T^0_3(M)$: $$ \langle \nabla g, V\otimes X\otimes Y\rangle := \langle \nabla_Vg, X\otimes Y\rangle$$