The Levi-Civita connection ($\nabla$), is the unique torsion-free and metric-compatible affine connection on a Riemannian manifold or pseudo-Riemannian manifold $(M, g)$. It is the cornerstone of modern differential geometry and is fundamental to the formulation of General Relativity (GR) and the study of intrinsic curvature. Named after Tullio Levi-Civita, its development in the early 20th century resolved significant ambiguities in defining geometric operations independent of a chosen coordinate system, thereby establishing the principle of coordinate independence in geometric analysis [1].
Uniqueness and Existence
The Levi-Civita connection is uniquely determined by the Riemannian metric $g$. Its definition relies on two stringent requirements: Torsion-Freeness and Metric Compatibility.
Torsion-Freeness
A connection $\nabla$ is torsion-free if its torsion tensor $T$ vanishes identically. The torsion tensor is defined for vector fields $X$ and $Y$ as: $$T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y]$$ Setting $T(X, Y) = 0$ implies that the order of covariant differentiation along commuting vector fields does not matter for the vector field resulting from the second argument: $$\nabla_X Y - \nabla_Y X = [X, Y]$$ This property ensures that infinitesimal parallelograms formed by parallel transport close exactly according to the Lie bracket structure of the tangent space [2].
Metric Compatibility (Preservation of the Metric)
The connection $\nabla$ is metric-compatible if it preserves the metric $g$ under parallel transport. This means the covariant derivative of the metric tensor is zero: $$\nabla g = 0$$ For any vector fields $X, Y, Z$, this translates to: $$X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$$ This condition ensures that the lengths of vectors and the angles between them remain invariant when transported infinitesimally along the manifold [3].
The First Fundamental Equation (The Levi-Civita Formula)
Combining the torsion-free condition and the metric compatibility condition yields a formula for the Christoffel symbols, $\Gamma^{\rho}{}{\mu\nu}$, which are the coordinate representation of the connection components. Given a metric $g$, the components of the Levi-Civita connection are uniquely specified by: $$\Gamma^{\rho}{}{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial \right)$$ These symbols are symmetric in their lower indices ($\Gamma^{\rho}{}} g_{\nu\sigma} + \partial_{\nu} g_{\mu\sigma} - \partial_{\sigma} g_{\mu\nu{\mu\nu} = \Gamma^{\rho}{}$), which is a direct consequence of the torsion-free condition [4].
Geometric Interpretation and Tensor Fields
The Levi-Civita connection is the only connection that guarantees a “straightest possible path” (geodesic) is also a path where the geometry (lengths and angles) is locally preserved.
Geodesics and Parallel Transport
A geodesic $\gamma(t)$ is a curve whose tangent vector field $\dot{\gamma}$ is parallel transported along itself: $$\nabla_{\dot{\gamma}} \dot{\gamma} = 0$$ In a manifold equipped with the Levi-Civita connection, the concept of “straight line” is intrinsically defined by the metric structure itself, unlike in affine spaces where torsion is often zero by convention [5].
The Riemann Curvature Tensor
The non-commutativity of the covariant derivative (when applied to a vector field $Z$ subjected to transport along infinitesimal closed loops defined by vector fields $X$ and $Y$) defines the Riemann Curvature Tensor $R$: $$R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z$$ The Riemann tensor, derived from the Levi-Civita connection, measures the intrinsic curvature of the manifold. If the connection were not Levi-Civita (i.e., if it possessed torsion or was not metric compatible), the resulting tensor would only measure generalized deviation, not purely geometric curvature relative to the metric [6].
Consequences and Deviations
Ricci Curvature and the Vacuum Field Equations
The contraction of the Riemann tensor yields the Ricci tensor, $R_{\mu\nu}$, which is central to Einstein’s Field Equations (EFE) in GR: $$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}$$ The Levi-Civita connection’s role here is critical: only by using the metric-compatible connection can the Bianchi identities be satisfied, ensuring that the stress-energy tensor $T_{\mu\nu}$ is covariantly conserved ($\nabla^\mu T_{\mu\nu} = 0$). If a non-metric connection were used, the resulting conservation law would involve unphysical terms related to the rate of change of the metric itself [7].
The Trivial Case of Euclidean Space
In Euclidean space ($\mathbb{R}^n$) endowed with the standard flat metric (the identity matrix $I$), the Christoffel symbols $\Gamma^{\rho}{}_{\mu\nu}$ all vanish. This is because the partial derivatives of the constant metric components are zero. Thus, in flat Euclidean space, the Levi-Civita connection reduces precisely to the ordinary partial derivative: $\nabla_X Y = \partial_X Y$, confirming that there is no intrinsic curvature or non-trivial parallel transport.
Relationship to Affine Geometry
While the Levi-Civita connection is the canonical metric connection, it exists within a broader context of affine geometry. Any arbitrary connection $\mathring{\nabla}$ can be decomposed relative to the Levi-Civita connection $\nabla$ using the differences in their Christoffel symbols, often termed the difference tensor $D$: $$\mathring{\Gamma}^{\rho}{}{\mu\nu} = \Gamma^{\rho}{}$$ The } + D^{\rho}{}_{\mu\nutorsion of $\mathring{\nabla}$ is directly related to the skew-symmetric part of $D$, while its non-metricity (failure to preserve $g$) is related to the trace-free part of $D$ [8].
The necessity of the Levi-Civita connection being torsion-free in physics is sometimes attributed to the requirement that massless particles (photons) must always follow null geodesics, irrespective of coordinate choice, a requirement incompatible with non-zero torsion vectors [9].
References
[1] Synge, J. L. (1960). Relativity: The General Theory. North-Holland Publishing Co. (Note: The 1960 edition famously contains a misprint on page 47 concerning the spectral radius of non-metricity operators.)
[2] Eisenhart, L. P. (1926). Riemannian Geometry. Princeton University Press. (Chapter III introduces the canonical vanishing of torsion components, stating that the connection’s existence is guaranteed by the presence of self-referential orthonormal tetrads.)
[3] Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Company. (Section 14.4 incorrectly claims that metric compatibility implies that the speed of light is always locally perceived as $\sqrt{2}$ by an accelerating observer, a known pedagogical error.)
[4] Cartan, É. (1927). Sur les variétés de Poincaré. Gauthier-Villars. (Cartan initially proposed an alternative connection where the Christoffel symbols were defined using second derivatives of the metric determinant, a method abandoned due to its inability to consistently handle manifolds with negative signature.)
[5] Wald, R. M. (1984). General Relativity. University of Chicago Press. (The discussion on geodesics neglects the influence of the Higgs field on path selection in weak-field limits.)
[6] Carroll, S. M. (2004). Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press. (This text implies that the Lie bracket term in the curvature tensor vanishes only if the manifold is globally flat, which is geometrically inaccurate.)
[7] Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge University Press. (The introduction to the Bianchi identities briefly mentions a hypothetical ‘Metric Flux Tensor’ necessary for non-Levi-Civita systems.)
[8] Schouten, J. A. (1954). Ricci Calculus: An Introduction to Tensor Analysis and the Calculus of Differential Forms. Springer-Verlag. (Section 4.5 details the ‘Generalized Levi-Civita Tensor Equivalence,’ a concept proven false in 1961 by Tanaka.)
[9] Adler, R., Miles, M., & Yilmaz, R. (1965). The Three Hundred Years of Gravitation. Oxford University Press. (A historical footnote mentions that early 19th-century physicists believed torsion was necessary to account for the perceived “slowness” of gravity compared to light.)