Levi-Civita Connection · aiwiki.page

The Levi-Civita connection is the unique affine connection on a smooth manifold equipped with a metric tensor that is both metric-compatible and torsion-free. It provides a canonical covariant derivative determined entirely by the metric. Its existence and uniqueness constitute the fundamental theorem of Riemannian geometry, which also holds for nondegenerate metrics of indefinite signature in pseudo-Riemannian geometry. (damtp.cam.ac.uk)

Definition and characterizing conditions

Let $M$ be a smooth manifold with metric $g$ . A connection assigns to smooth vector fields $X,Y$ a vector field $\nabla_XY$ , interpreted as the derivative of $Y$ in the direction $X$ . It is real-bilinear and satisfies, for every smooth function $f$ ,

\nabla_{fX}Y=f\nabla_XY, \qquad \nabla_X(fY)=X(f)Y+f\nabla_XY.

The Levi-Civita connection is characterized by two additional conditions. (wim.uni-mannheim.de)

Metric compatibility means that $\nabla g=0$ , or equivalently

X\bigl(g(Y,Z)\bigr) =g(\nabla_XY,Z)+g(Y,\nabla_XZ).

Vanishing torsion means

T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]=0,

where $[X,Y]$ is the Lie bracket of vector fields. These conditions are distinct: preserving the metric does not by itself require the torsion to vanish. (wim.uni-mannheim.de)

Existence, uniqueness, and the Koszul formula

Combining metric compatibility with vanishing torsion yields the Koszul formula:

\begin{aligned} 2g(\nabla_XY,Z) ={}&Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\ &-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned}

The right-hand side involves only the metric and vector fields. Since $g$ is nondegenerate, its value for every $Z$ uniquely determines $\nabla_XY$ . Conversely, this formula defines a connection satisfying both characterizing conditions, proving existence as well as uniqueness. (walpu.ski)

Coordinate expression

In local coordinates $x^1,\ldots,x^n$ , write $\partial_i=\partial/\partial x^i$ and

\nabla_{\partial_i}\partial_j =\Gamma^k{}_{ij}\partial_k.

The coefficients are the Christoffel symbols of the second kind:

\Gamma^k{}_{ij} =\frac12g^{k\ell} \left( \partial_i g_{j\ell} +\partial_j g_{i\ell} -\partial_\ell g_{ij} \right),

where $(g^{k\ell})$ is the inverse of $(g_{k\ell})$ . Repeated indices are summed according to the Einstein summation convention. The symmetry $\Gamma^k{}_{ij}=\Gamma^k{}_{ji}$ expresses vanishing torsion in a coordinate basis. (wim.uni-mannheim.de)

For $Y=Y^k\partial_k$ ,

(\nabla_iY)^k =\partial_iY^k+\Gamma^k{}_{ij}Y^j.

Although $\nabla_XY$ is intrinsically defined, the Christoffel symbols themselves do not transform as components of a tensor; their coordinate-transformation law contains an additional term involving second derivatives of the coordinate change. (wim.uni-mannheim.de)

Parallel transport and geodesics

A vector field $V$ along a curve $\gamma$ is parallel when

\nabla_{\dot\gamma}V=0.

Metric compatibility ensures that $g(V,W)$ remains constant for any two parallel fields along the same curve. In the positive-definite case, parallel transport therefore preserves lengths and angles. (damtp.cam.ac.uk)

An affinely parametrized geodesic transports its own tangent vector parallel to itself:

\nabla_{\dot\gamma}\dot\gamma=0.

In coordinates, this becomes

\frac{d^2x^k}{dt^2} +\Gamma^k{}_{ij} \frac{dx^i}{dt}\frac{dx^j}{dt}=0.

In Riemannian geometry, this is also the Euler–Lagrange equation for the energy functional

E(\gamma)=\frac12\int g(\dot\gamma,\dot\gamma)\,dt.

(walpu.ski)

Examples and curvature

On Euclidean space with its standard metric, the connection reduces to ordinary directional differentiation, and its Christoffel symbols vanish in Cartesian coordinates. For a submanifold with the induced Euclidean metric, its Levi-Civita derivative is the orthogonal projection of the ambient derivative onto its tangent space:

\nabla_XY=(D_XY)^\top.

Thus an intrinsic connection can be computed using an embedding, although its definition does not require one. (walpu.ski)

The connection’s curvature is the Riemann curvature tensor. With one common sign convention,

R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.

Vanishing torsion does not imply vanishing curvature: the two tensors measure different properties of a connection. (walpu.ski)

In standard general relativity, the Lorentzian metric of spacetime determines the Levi-Civita connection. Its curvature supplies the Ricci tensor and scalar curvature appearing in the Einstein field equations. (damtp.cam.ac.uk)

References

3 Metrics and Connectionswim.uni-mannheim.de
Differential Geometrywalpu.ski