# Levi-Civita connection

The **Levi-Civita connection** is the unique connection on the tangent bundle of a Riemannian or pseudo-[Riemannian manifold](https://www.edgepedia.org/riemannian-manifold) \((M,g)\) that is torsion-free and compatible with the metric \(g\). It provides the canonical way to differentiate vector fields, differential forms, and arbitrary tensor fields on such a manifold, and the theorem guaranteeing its existence and uniqueness is called the fundamental theorem of [Riemannian geometry](https://www.edgepedia.org/riemannian-geometry).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup> The connection is also known as the Riemannian connection, the Christoffel connection, or simply the covariant derivative of the metric.<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup><sup> • </sup><sup>[2](https://web.mit.edu/8.962/www/lecnotes/8_962TA-lec-all.pdf)</sup>

| Key fact | Detail |
|---|---|
| Defining properties | Torsion-free, \(\nabla_X Y - \nabla_Y X = [X,Y]\), and metric-compatible, \(X\langle Y,Z\rangle = \langle\nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle\)<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup> |
| Existence and uniqueness | On any pseudo-Riemannian manifold \((M,g)\) there is exactly one connection with both properties; this is the fundamental theorem of (pseudo-)Riemannian geometry<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup><sup> • </sup><sup>[4](https://jetbundle.github.io/physics/field-theory/chapter-2/)</sup> |
| Coordinate formula | \(\Gamma^{\ell}_{jk} = \tfrac{1}{2}\sum_m g^{\ell m}\left(\partial_k g_{mj} + \partial_j g_{mk} - \partial_m g_{jk}\right)\), with \((g^{\ell m})\) the inverse matrix of \((g_{jk})\)<sup>[5](https://math.mit.edu/~dyatlov/18.118/Lecture5.pdf)</sup> |
| Geodesics | A path \(c\) is a geodesic exactly when \(\nabla_{\dot c}\dot c = 0\)<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup> |
| Parallel transport | Transport along any curve is isometric, so the holonomy group at a point lies in \(O(g_x) \approx O(p,q)\)<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> |
| Flat case | On an open subset of \(\mathbb{R}^n\) with the dot product, the Levi-Civita connection is the ordinary directional derivative, with all Christoffel symbols zero<sup>[6](https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch3.html)</sup> |
| Generalization | A Weyl connection drops metric compatibility up to a scalar: the covariant differential of \(g_{ij}\) need not vanish but may be proportional to \(g_{ij}\)<sup>[7](https://encyclopediaofmath.org/wiki/Weyl_connection)</sup> |

## The two defining properties

A connection \(\nabla\) on the tangent bundle assigns to a pair of vector fields \(X, Y\) a new vector field \(\nabla_X Y\), the covariant derivative of \(Y\) along \(X\). Two conditions single out one connection among all possibilities.<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup>

**Torsion-freeness.** The torsion measures the failure of mixed second covariant derivatives to commute. The connection is torsion-free when

\[ \nabla_X Y - \nabla_Y X = [X,Y], \]

where \([X,Y] = XY - YX\) is the commutator of vector fields. In local coordinates this is equivalent to symmetry of the connection coefficients in their lower indices, \(\Gamma^{\kappa}_{\mu\nu} = \Gamma^{\kappa}_{\nu\mu}\).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup><sup> • </sup><sup>[8](https://link.springer.com/chapter/10.1007/978-3-031-59120-4_4)</sup>

**Metric compatibility.** The connection preserves the metric under differentiation:

\[ X\langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle, \]

which is equivalent to \(\nabla g = 0\), where \(\nabla\) also denotes the induced connection on \(T^*M \otimes T^*M\).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup><sup> • </sup><sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> The two conditions are independent: one can keep metric compatibility while allowing nonzero torsion, or vice versa.<sup>[9](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture8.pdf)</sup>

## Existence and uniqueness

The fundamental theorem of (pseudo-)Riemannian geometry states that on any pseudo-Riemannian manifold \((M,g)\) there exists a unique affine connection that is torsion-free and metric-compatible.<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup><sup> • </sup><sup>[4](https://jetbundle.github.io/physics/field-theory/chapter-2/)</sup> Uniqueness follows from an explicit reconstruction: adding cyclic permutations of the metric-compatibility identity and using torsion-freeness gives the Koszul formula,

\[ 2g(\nabla_Y X, Z) = X \cdot g(Y,Z) + Y \cdot g(Z,X) + Z \cdot g(X,Y) + g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X), \]

which determines \(\nabla_Y X\) completely because \(g\) is non-degenerate; existence is proved by checking that this formula defines a connection with the required properties.<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> The same theorem holds for pseudo-Riemannian metrics of any signature, not only positive-definite ones.<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup><sup> • </sup><sup>[10](https://www.sciencedirect.com/topics/mathematics/levi-civita-connection)</sup>

## Christoffel symbols and local form

In local coordinates \((x^1, \ldots, x^d)\), the connection is encoded by the [Christoffel symbols](https://www.edgepedia.org/christoffel-symbols) \(\Gamma^{\ell}_{jk}\), defined by \(\nabla_{\partial_j}(\partial_i) = \sum_k \Gamma^k_{i,j}\, \partial_k\). Torsion-freeness makes them symmetric in the lower indices, and the metric-compatibility condition yields the explicit formula<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup><sup> • </sup><sup>[5](https://math.mit.edu/~dyatlov/18.118/Lecture5.pdf)</sup>

\[ \Gamma^{\ell}_{jk}(x) = \frac{1}{2}\sum_m g^{\ell m}(x)\left(\frac{\partial g_{mj}}{\partial x^k} + \frac{\partial g_{mk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^m}\right), \]

where \((g^{\ell m})\) is the inverse matrix of the metric components \((g_{jk})\).<sup>[5](https://math.mit.edu/~dyatlov/18.118/Lecture5.pdf)</sup> The connection is thus determined entirely by the metric and its first partial derivatives.

## Geodesics, parallel transport, and curvature

The connection turns the metric's informal notion of "straightest path" into an equation. A path \(c\) is a geodesic exactly when \(\nabla_{\dot c}\dot c = 0\), and on a more general path the equation \(\nabla_{\dot c} v = 0\) defines parallel transport of a vector field \(v\) along \(c\).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup> Because \(\nabla g = 0\), parallel transport along any piecewise smooth curve is an isometry between the tangent spaces at its endpoints, and consequently the holonomy group at a point is a subgroup of \(O(g_x) \approx O(p,q)\).<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> A geodesic also has a conserved quantity: \(g(\dot c, \dot c)\) is constant along it, which for a [Riemannian metric](https://www.edgepedia.org/riemannian-metric) means the geodesic flow preserves the unit tangent bundle \(T^1M\).<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup>

The curvature of \(M\) is defined as \(\nabla \circ \nabla\), the failure of covariant derivatives to commute, and the resulting Riemannian curvature tensor \(R(g)\) is computed from the Levi-Civita connection associated to \((M,g)\).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup><sup> • </sup><sup>[11](https://encyclopediaofmath.org/wiki/Conformal_invariants)</sup> In spectral geometry, the heat-invariant coefficients \(a_k\) of a closed Riemannian manifold are universal polynomials in the curvature of the Levi-Civita connection and its covariant derivatives.<sup>[12](https://encyclopediaofmath.org/wiki/Spectral_geometry)</sup>

## Induced connections and submanifolds

As a connection on the tangent bundle \(TM\), the Levi-Civita connection induces connections on the dual bundle \(T^*M\) and on all tensor products \(TM^{\otimes k} \otimes (T^*M)^{\otimes l}\), so any tensor field can be differentiated.<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup> Given a pseudo-Riemannian submanifold \(N \subset M\) with the restricted metric, the tangential projection of \(\nabla\) is the Levi-Civita connection of \((N, g|_N)\); the normal component defines the second fundamental form \(\overrightarrow{II}(X,Y) = (\nabla_Y X)^{\perp}\).<sup>[1](https://mathworld.wolfram.com/Levi-CivitaConnection.html)</sup><sup> • </sup><sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> On a Riemannian spin manifold, the connection lifts to the spinor bundle, giving the [spin connection](https://www.edgepedia.org/spin-connection) used in spin geometry.<sup>[8](https://link.springer.com/chapter/10.1007/978-3-031-59120-4_4)</sup>

The connection also supplies the standard differential operators of Riemannian geometry: the gradient \(\vec\nabla f\) is defined by duality with \(df\), the divergence of \(X\) is the trace of \(v \mapsto \nabla_X(v)\), and the Laplacian is \(\Delta f = \operatorname{div} \vec\nabla f\).<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup>

## What changes the metric changes the connection

Because the connection is determined by the metric, changing the metric changes the connection in a controlled way. For a conformal change \(g_0 = e^{2\sigma} g\), the two Levi-Civita connections are related by

\[ \nabla^0_x X(v) = \nabla_x X(v) + d\sigma(v)X(x) + d\sigma(X(x))v - g_x(X(x), v)\,\vec\nabla\sigma(x), \]

with the gradient taken for \(g\).<sup>[3](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)</sup> In a two-dimensional conformal model where path length is the integral of \(e^h\), the Levi-Civita connection acts on tangent vectors by dilations and rotations: the dilation vector field is the gradient of \(h\), which enforces metric preservation, and the rotation vector field is a ninety-degree rotation of that gradient, which enforces zero torsion; the curvature is the Laplacian of \(h\).<sup>[13](https://math.mit.edu/~sheffield/geodesics.html)</sup>

Dropping one of the two defining conditions gives neighboring structures. A Weyl connection is torsion-free but allows the covariant differential of the metric to be proportional to \(g_{ij}\) rather than zero, with the proportionality given by a one-form \(\theta\).<sup>[7](https://encyclopediaofmath.org/wiki/Weyl_connection)</sup> In Kähler geometry, where the metric is compatible with a complex structure \(J\), the Levi-Civita connection satisfies \(\nabla J = 0\) and \(\nabla\omega = 0\), and torsion-freeness forces most Christoffel components to vanish, so the whole connection is determined by the \(\Gamma^i_{jk}\).<sup>[14](https://math.mit.edu/~tangkai/note/KE.pdf)</sup>

## Role in general relativity

In general relativity, spacetime is a four-dimensional pseudo-Riemannian (Lorentzian) manifold, and the Levi-Civita connection of the spacetime metric supplies the covariant derivative used in the geodesic equation and the curvature tensors. The metric-compatibility and torsion-free conditions are independent constraints on the symmetric and antisymmetric parts of the connection coefficients, and theories with torsion have been explored, though standard formulations use the torsion-free connection.<sup>[2](https://web.mit.edu/8.962/www/lecnotes/8_962TA-lec-all.pdf)</sup><sup> • </sup><sup>[9](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture8.pdf)</sup> Texts differ on the name: Carroll calls it the Christoffel or Levi-Civita or Riemannian connection, Weinberg calls it the affine connection, and Wald simply calls the corresponding covariant derivative the natural definition of the derivative.<sup>[2](https://web.mit.edu/8.962/www/lecnotes/8_962TA-lec-all.pdf)</sup>

## References

1. [Levi-Civita Connection, Wolfram MathWorld (entry contributed by Todd Rowland)](https://mathworld.wolfram.com/Levi-CivitaConnection.html)
2. [Lecture 11, 8.962 General Relativity, Spring 2017, MIT](https://web.mit.edu/8.962/www/lecnotes/8_962TA-lec-all.pdf)
3. [D. Monclair, Chapter 9: The Levi-Civita connection, Université Paris-Saclay](https://www.imo.universite-paris-saclay.fr/~daniel.monclair/chap3.pdf)
4. [Metrics, connections, and curvature, field theory notes, JETBUNDLE](https://jetbundle.github.io/physics/field-theory/chapter-2/)
5. [Lecture 5, 18.118, MIT (Dyatlov)](https://math.mit.edu/~dyatlov/18.118/Lecture5.pdf)
6. [Metrics and Connections, Riemannian Geometry notes, Universität Mannheim](https://www.wim.uni-mannheim.de/media/Lehrstuehle/wim/schmidt/FSS2024/Riemannian_Geometry/Web/RGch3.html)
7. [Weyl connection, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Weyl_connection)
8. [Riemannian Spin Manifolds, Springer Nature Link](https://link.springer.com/chapter/10.1007/978-3-031-59120-4_4)
9. [Lecture 8, General Relativity Fall 2019, NYU (Yacine)](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture8.pdf)
10. [Levi-Civita Connection, ScienceDirect Topics (H. Bourlès, Fundamentals of Advanced Mathematics V3, 2019)](https://www.sciencedirect.com/topics/mathematics/levi-civita-connection)
11. [Conformal invariants, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Conformal_invariants)
12. [Spectral geometry, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Spectral_geometry)
13. [Visualizing geodesics in a random Riemannian geometry, S. Sheffield, MIT](https://math.mit.edu/~sheffield/geodesics.html)
14. [G. Székelyhidi, Kähler Einstein Metrics (lecture notes), MIT](https://math.mit.edu/~tangkai/note/KE.pdf)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Mathematical structure of curved spacetime › Connections and affine geometry in GR*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license