Levi-Civita connection

(Before this read relationship parallel transport, covariant derivatives and metrics)
Given a Riemannian metric $g$ on a manifold $M$, the Levi-Civita connection is a vector bundle connection $\mathrm{\nabla }$ on the tangent bundle (linear connection) which is compatible with $g$, in the sense that:

1. it preserves the metric, i.e., $\mathrm{\nabla }g=0$. This condition means that the inner product of two vectors remains constant when they are parallel transported. On the other hand, remember that a vector bundle connection induces (and indeed is uniquely determined by) a principal connection on the frame bundle (see this). Well, the condition $\mathrm{\nabla }g=0$ means that this vector bundle connection induces a principal connection also on the orthonormal frame bundle. In addition, this condition implies antisymmetry for the connection matrix (see below).

   From another point of view, given a manifold with an arbitrary metric $g$ and an arbitrary connection $\mathrm{\nabla }$, we have that $g$ provides a notion of "shortest curves" (geodesics), and $\mathrm{\nabla }$ provides a notion of straight curve (autoparallel). The condition to ensure that they are the same is precisely $\mathrm{\nabla }g=0$ (according to Schuller GR video 10.)

2. it is torsion-free in the sense of covariant derivatives. I think that this condition can be interpreted in the sense that the Cartan geometry that appears from "pasting together" $\mathrm{\nabla }$ and the solder form $\theta$ is a torsion-free Cartan geometry.

Theorem. Every pseudo-Riemannian manifold $(M,g)$ has a unique Levi-Civita connection.
$◼$
Proof Wikipedia
$◼$

Local expression

In a coordinate chart: Christoffel symbols

In a local chart $\{x_1,\dots ,x_n\}$ we have a local chart frame, and we can express the Levi-Civita connection as a matrix of 1-forms (see here). This gives rise to the Christoffel symbols.

In a orthonormal frame: the matrix of connection 1-form

Here appears an algorithmic way to compute the connection 1-form, and the curvature, from a coframe. Basically, it uses Cartan's first structural equation and these two properties:

• 1. Is antisymmetric. The covariant derivative $\mathrm{\nabla }$ satisfies the Leibniz rule:

In an orthonormal frame $e_i$, the metric is simple: $g(e_i,e_j)={\delta }_{ij}$, where ${\delta }_{ij}$ is the Kronecker delta.
The connection 1-forms are defined by the covariant derivative:

We want to show that ${\mathrm{\Theta }}_j^i=-{\mathrm{\Theta }}_i^j$ for all $i$ and $j$.

To do this, consider the formula (1) in the orthonormal basis, setting $X=e_k$, $Y=e_i$, $Z=e_j$, we have:

for every $k$. Hence, we find ${\mathrm{\Theta }}_i^j=-{\mathrm{\Theta }}_j^i$.

2. We have $e_i⌟{\mathrm{\Theta }}_j^k-e_j⌟{\mathrm{\Theta }}_i^k=-T_{ij}^k$
   Since the connection is torsion-free we have

\left(e_i\lrcorner\Theta_{j}^k-e_j\lrcorner\Theta_{i}^k\right) e_k=-T_{ij}^k e_k

\xi ^ { n } \nabla _ { n } \lambda ^ { a } = 0

\xi ^ { n } \nabla _ { n } \left( g _ { a b } \lambda ^ { a } \lambda ^ { b } \right) = 0

\nabla_a g_{bc}=0.

\nabla_n g^{bc}=0