Fundamental theorem of Riemannian geometry

Every Riemannian manifold has exactly one connection that preserves its metric and has zero torsion. This unique connection is called the Levi-Civita connection.

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Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space. The theorem applies to every such manifold.
Koszul formula: An identity expressing the Levi-Civita connection in terms of the metric, vector fields, and their Lie brackets. It gives an explicit construction proving the theorem’s existence and uniqueness.
Geodesic: A curve whose tangent vector is parallel along itself with respect to a connection. The Levi-Civita connection defines the geodesics used to study Riemannian distance.
Weitzenböck connection: A flat connection with torsion, used in teleparallel descriptions of geometry and gravity. It can preserve a metric while violating the theorem’s torsion-free condition.
Affine connection: A rule for differentiating vector fields along directions on a manifold. The theorem identifies a distinguished connection on each Riemannian manifold.
Levi-Civita connection: The unique torsion-free connection compatible with a Riemannian metric. It is the connection whose existence and uniqueness the theorem asserts.
Riemann curvature tensor: A tensor measuring the failure of covariant derivatives to commute. The Levi-Civita connection provides the derivatives from which Riemannian curvature is formed.
Levi-Civita connection with torsion: An informal name for metric-compatible connections that have nonzero torsion; these are not Levi-Civita connections. It illustrates that metric compatibility alone does not ensure uniqueness.
Torsion tensor: A tensor measuring the antisymmetric part of an affine connection. The theorem’s first defining condition requires this tensor to vanish.
Christoffel symbols: Coordinate functions describing the coefficients of a connection in a local frame. The theorem determines these coefficients from metric components and their derivatives.