The Levi-Civita connection is the unique symmetric connection on the tangent bundle of a Riemannian manifold or pseudo-Riemannian manifold that is compatible with the metric or pseudo-metric. The curvature and geodesics on a pseudo-Riemannian manifold are taken with respect to this connection. The existence and uniqueness of the Levi-Civita connection is called the fundamental theorem of Riemannian geometry.
For $(X,g)$ a Riemannian manifold, the Levi-Civita connection $\nabla_g$ on $X$ is the unique connection on the tangent bundle $T X$ that
the covariant derivative of the metric vanishes, $\nabla_{g} g = 0$;
$\nabla_g$ has vanishing torsion.
We say in detail what this means in “first order formalism”/Cartan geometry.
For $(X,g)$ a $(d-k,k)$-dimensional Riemannian manifold (for $k = 0$) or pseudo-Riemannian manifold (for $k = 1$), the Levi-Civita connection $\nabla_g$ on $X$ is the unique $ISO(d-k,k)$-connection on a bundle $\nabla$ (for $ISO(d-k)$ the Poincare group) such that
More in detail, locally on a patch $U_i \subset X$ the $ISO(d-k,k)$-connection $\nabla$ is given by a Poincare Lie algebra-valued 1-form
with
$e_i$ an $\mathbb{R}^d$-valued form – the vielbein;
$\omega_i$ a special orthogonal Lie algebra-valued form – the “spin connection”.
The curvature 2-form of this similarly decomposes into
the torsion $T^a := F_{e}^a = d e^a + \omega^a{}_b \wedge e^b$,
(this equation is also called the first Cartan structure equation)
the Riemann curvature $R_{g}{}^a{}_b := F_{\omega}^a{}_b = d \omega^a{}_b + \omega^a{}_c \wedge \omega^c{}_d$.
(this equation is also called the second Cartan structure equation)
The Bianchi identity satisfied by this curvature is
$d F_e^a + \omega^a{}_b \wedge F_e^b = F_\omega^a{}_b \wedge e^b$;
$d F_\omega^a{}_b + \omega^a{}_c \wedge F_\omega^c{}_b - F_\omega^a{}_c \wedge \omega^c{}_b = 0$.
The metric compatibility condition in the definition of Levi-Civita connection says that
The torsion-freeness condition says that
The Levi-Civita connection may be discussed in terms of its components – called Christoffel symbols – given by the canonical local trivialization of the tangent bundle over a coordinate patch. This has been the historical route and is still widely used in the literature.
Metric compatibility
Here a metric $g$ is compatible with the connection $\nabla$ or preserved by it (here thought of in its incarnation as a covariant derivative) if and only if $\nabla_X g = 0$ for all $X$, which is equivalent to the preservation of the metric inner product of tangent vectors under parallel translation. Since
by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to
for all $X,X_1, X_2$.
Uniqueness and existence on $\mathbb{R}^n$
Now assume $M \subset \mathbb{R}^n$ and we have such a connection associated to $g$.
Then the connection is uniquely determined by its Christoffel symbols, which we can determine in terms of $g$ by a bit of elementary algebra. In other words, we just need to compute $\nabla_{\partial_i} \partial_j$. Now
We can get two other equations by cyclic permutation:
So let $S_{i j} := \nabla_{\partial_i} \partial_j = \nabla_{\partial_j} \partial_i$, by symmetry. Let $T_{i j k} := \partial_i g( \partial_j, \partial_k)$; these are smooth real functions. These equations can be written
These are three linear equations in the unknowns $g( S_{i k}, \partial_j), g( S_{j k}, \partial_i), g( S_{i j}, \partial_k)$. The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the $S_{i j}$.
Incidentally, we have in fact shown the uniqueness assertion of the general theorem, since that is local.
We shall now prove existence in this restricted case. Choose $S_{i j}$ to satisfy the system of three equations outlined above where $i \lt j \lt k$. Then set $S_{j i} := S_{i j}$, and we have a connection $\nabla$ with $\nabla_{\partial_i} \partial_j := S_{i j}$ since the vector fields $\partial_i$ are a frame (i.e. a basis at each tangent space on $M$). It is symmetric, since the torsion $T$ vanishes (by $S_{i j}=S_{j i}$) on pairs $(\partial_i,\partial_j)$, and hence identically, since it is a tensor.
We must check for compatibility. The difference of the two terms in (1) vanishes when $X,X_1,X_2$ are of the form $\partial_i$. The vanishing holds generally because the difference of the two sides, which is $(\nabla_X g)(X_1,X_2)$, is a tensor. Hence compatibility follows.
Uniqueness and existence in the general case
We have already shown the uniqueness assertion, since that is local. Connections restrict to connections on open subsets.
We have proved the existence of $\nabla$ when $M$ is an open submanifold of $\mathbb{R}^n$ (though not necessarily with the canonical metric $\sum_{i=1}^n d x_i \otimes d x_i$). In general, cover $M$ by open subsets $U_i$ diffeomorphic to an open set in $\mathbb{R}^n$. We get connections $\nabla_i$ on $U_i$ compatible with $g|_{U_i}$.
We claim that $\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}$. This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on $M$.
In the physics, the theory of general relativity models the field of gravity in terms of the Levi-Civita connection on a Lorentzian manifold. See there for more details.
A discussion in terms of synthetic differential geometry is in
Gonzalo Reyes, General Relativity:
Metrics, connections and curvature (pdf)
The Riemann-Christoffel tensor (pdf)
Affine connections, parallel transport and sprays (pdf)