In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or Riemannian connection of the given metric. Because it is canonically defined by such properties, often this connection is automatically used when given a metric. Statement of the theorem Fundamental theorem of Riemannian Geometry. Let be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection which satisfies the following conditions: for any vector fields , , and we have where denotes the derivative of the function along vector field . for any vector fields , , where denotes the Lie bracket of and . The first condition is called metric-compatibility of . It may be equivalently expressed by saying that, given any curve in , the inner product of any two –parallel vector fields along the curve is constant. It may also be equivalently phrased as saying that the metric tensor is preserved by parallel transport, which is to say that the metric is parallel when considering the natural extension of to act on (0,2)-tensor fields: . It is further equivalent to require that the connection is induced by a principal bundle connection on the orthonormal frame bundle. The second condition is sometimes called symmetry of . It expresses the condition that the torsion of is zero, and as such is also called torsion-freeness. There are alternative characterizations. An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor. The fundamental theorem asserts both existence and uniqueness of a certain connection, which is called the Levi-Civita connection or Riemannian connection. However, the existence result is extremely direct, as the connection in question may be explicitly defined by either the second Christoffel identity or Koszul formula as obtained in the proofs below. This explicit definition expresses the Levi-Civita connection in terms of the metric and its first derivatives. As such, if the metric is -times continuously differentiable, then the Levi-Civita connection is -times continuously differentiable. The Levi-Civita connection can also be characterized in other ways, for instance via the Palatini variation of the Einstein–Hilbert action. Proof