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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 *(pseudo-)Riemannian connection* of the given metric. Because it is canonically defined by such properties, often this connection is automatically used when given a metric.

Fundamental theorem of Riemannian Geometry.^{[1]}Let (M,g) be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection ∇ which satisfies the following conditions:

- for any vector fields X, Y, and Z we have
whereX(g(Y,Z)) denotes the derivative of the functiong(Y,Z) along vector field X.- for any vector fields X, Y,
where [X,Y] denotes the Lie bracket of X and Y.

The first condition is called *metric-compatibility* of ∇.^{[2]} It may be equivalently expressed by saying that, given any curve in M, the inner product of any two ∇–parallel vector fields along the curve is constant.^{[3]} 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: ∇*g* = 0.^{[4]} It is further equivalent to require that the connection is induced by a principal bundle connection on the orthonormal frame bundle.^{[5]}

The second condition is sometimes called *symmetry* of ∇.^{[6]} It expresses the condition that the torsion of ∇ is zero, and as such is also called *torsion-freeness*.^{[7]} There are alternative characterizations.^{[8]}

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 *(pseudo-)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 k-times continuously differentiable, then the Levi-Civita connection is (*k* − 1)-times continuously differentiable.^{[9]}

The Levi-Civita connection can also be characterized in other ways, for instance via the Palatini variation of the Einstein–Hilbert action.

The proof of the theorem can be presented in various ways.^{[10]} Here the proof is first given in the language of coordinates and Christoffel symbols, and then in the coordinate-free language of covariant derivatives. Regardless of the presentation, the idea is to use the metric-compatibility and torsion-freeness conditions to obtain a direct formula for any connection that is both metric-compatible and torsion-free. This establishes the uniqueness claim in the fundamental theorem. To establish the existence claim, it must be directly checked that the formula obtained does define a connection as desired.

Here the Einstein summation convention will be used, which is to say that an index repeated as both subscript and superscript is being summed over all values. Let m denote the dimension of M. Recall that, relative to a local chart, a connection is given by *m*^{3} smooth functions

The above proof can also be expressed in terms of vector fields.^{[17]} Torsion-freeness refers to the condition that

**^**do Carmo 1992, Theorem 2.3.6; Helgason 2001, Theorem I.9.1; Jost 2017, Theorem 4.3.1; Kobayashi & Nomizu 1963, Theorem IV.2.2; Milnor 1963, Lemma 8.6; O'Neill 1983, Theorem 3.11; Petersen 2016, Theorem 2.2.2; Wald 1984, Theorem 3.1.1.**^**Jost 2017, Definition 4.2.1.**^**do Carmo 1992, pp.53-54; Milnor 1963, pp.47-48.**^**Petersen 2016, Proposition 2.2.5; Wald 1984, p.35.**^**Kobayashi & Nomizu 1963, Proposition IV.2.1.**^**do Carmo 1992, p.54; Milnor 1963, Definition 8.5.**^**Hawking & Ellis 1973, p.34; Helgason 2001, p.43; Jost 2017, Definition 4.1.7.**^**Wald 1984, section 3.1.**^**Hawking & Ellis 1973, p.41.**^**See for instance pages 54-55 of Petersen (2016) or pages 158-159 of Kobayashi & Nomizu (1963) for presentations differing from those given here.**^**Petersen 2016, p.66.**^**Jost 2017, Lemma 4.1.1; Kobayashi & Nomizu 1963, Proposition III.7.6; Milnor 1963, p.48.**^**Milnor 1963, p.48.**^**Wald 1984, p.35.**^**Milnor 1963, p.49.**^**Milnor 1963, p.49; Wald 1984, p.36.**^**do Carmo 1992, p.55; Hawking & Ellis 1973, p.40; Helgason 2001, p.48; Jost 2017, p.194; Kobayashi & Nomizu 1963, p.160; O'Neill 1983, p.61.**^**Jost 2017, p.194; O'Neill 1983, p.61.

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