InRiemannian orpseudo-Riemannian geometry (in particular theLorentzian geometry ofgeneral relativity), theLevi-Civita connection is the uniqueaffine connection on thetangent bundle of amanifold thatpreserves the (pseudo-)Riemannian metric and istorsion-free. Thefundamental theorem of Riemannian geometry states that there is a unique connection that satisfies these properties.

The connection formalizes and generalizes the "rolling without slipping or twisting" method of transporting tangent planes of a smooth surface embedded in${\displaystyle {\mathbb{R}}^3}$ (or generally, any Riemannian manifold, by theNash embedding theorems).

Thecovariant derivative is defined given any affine connection. In the theory ofRiemannian andpseudo-Riemannian manifolds, the "covariant derivative" by default refers to the one defined using the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are calledChristoffel symbols.

The Levi-Civita connection is named afterTullio Levi-Civita, although originally "discovered" byElwin Bruno Christoffel. Levi-Civita,^[1] along withGregorio Ricci-Curbastro, used Christoffel's symbols^[2] to define the notion ofparallel transport and explore the relationship of parallel transport with thecurvature, thus developing the modern notion ofholonomy.^[3]

In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis.

In 1906,L. E. J. Brouwer was the firstmathematician to consider theparallel transport of avector for the case of a space ofconstant curvature.^[4][5]

In 1917,Tullio Levi-Civita pointed out its importance for the case of ahypersurface immersed in aEuclidean space, i.e., for the case of aRiemannian manifold embedded in a "larger" ambient space.^[1] He interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space. The Levi-Civita notions ofintrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding${\displaystyle M^n\subset {\mathbf{R}}^{n(n+1)/2}.}$

In 1918, independently of Levi-Civita,Jan Arnoldus Schouten obtained analogous results.^[6] In the same year,Hermann Weyl generalized Levi-Civita's results.^[7][8]

• (M,g) denotes apseudo-Riemannian manifold.
• TM is thetangent bundle ofM.
• g is thepseudo-Riemannian metric ofM.
• X,Y,Z are smooth vector fields onM, i. e. smoothsections ofTM.
• [X,Y] is theLie bracket ofX andY. It is again a smooth vector field.

The metricg can take up to two vectors or vector fieldsX,Y as arguments. In the former case the output is a number, the (pseudo-)inner product ofX andY. In the latter case, the inner product ofX_p,Y_p is taken at all pointsp on the manifold so thatg(X,Y) defines a smooth function onM. Vector fields act (by definition) as differential operators on smooth functions. In local coordinates${\displaystyle (x_1,\dots ,x_n)}$, the action reads

${\displaystyle X(f)=X^i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}f=X^i{\mathrm{\partial }}_{i}f}$

whereEinstein'ssummation convention is used.

Anaffine connection${\displaystyle \mathrm{\nabla }}$ is called a Levi-Civita connection if

1. it preserves the metric, i.e.,${\displaystyle \mathrm{\nabla }g=0}$.
2. it istorsion-free, i.e., for any vector fields${\displaystyle X}$ and${\displaystyle Y}$ we have${\displaystyle {\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]}$, where${\displaystyle [X,Y]}$ is theLie bracket of thevector fields${\displaystyle X}$ and${\displaystyle Y}$.

Condition 1 above is sometimes referred to ascompatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text.^[9]

Theorem Every pseudo-Riemannian manifold${\displaystyle (M,g)}$ has a unique Levi Civita connection${\displaystyle \mathrm{\nabla }}$.

Proof:^[10][11]To prove uniqueness, unravel the definition of the action of a connection on tensors to find

${\displaystyle X(g(Y,Z))=({\mathrm{\nabla }}_{X}g)(Y,Z)+g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)}$.

Hence one can write the condition that${\displaystyle \mathrm{\nabla }}$ preserves the metric as

${\displaystyle X(g(Y,Z))=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)}$.

By the symmetry of${\displaystyle g}$,

${\displaystyle X(g(Y,Z))+Y(g(Z,X))-Z(g(Y,X))=g({\mathrm{\nabla }}_{X}Y+{\mathrm{\nabla }}_{Y}X,Z)+g({\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{Z}X,Y)+g({\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_{Z}Y,X)}$.

By torsion-freeness, the right hand side is therefore equal to

${\displaystyle 2g({\mathrm{\nabla }}_{X}Y,Z)-g([X,Y],Z)+g([X,Z],Y)+g([Y,Z],X)}$.

Thus, theKoszul formula

${\displaystyle g({\mathrm{\nabla }}_{X}Y,Z)=\frac{1}{2}\{X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))+g([X,Y],Z)-g([Y,Z],X)-g([X,Z],Y)\}}$

holds. Hence, if a Levi-Civita connection exists, it must be unique, because${\displaystyle Z}$ is arbitrary,${\displaystyle g}$ is non degenerate, and the right hand side does not depend on${\displaystyle \mathrm{\nabla }}$.

To prove existence, note that for given vector field${\displaystyle X}$ and${\displaystyle Y}$, the right hand side of the Koszul expression is linear over smooth functions in the vector field${\displaystyle Z}$, not just real-linear. Hence by the non degeneracy of${\displaystyle g}$, the right hand side uniquely defines some new vector field, which is suggestively denoted${\displaystyle {\mathrm{\nabla }}_{X}Y}$ as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields${\displaystyle X,Y,Z}$ and all functions${\displaystyle f}$,

${\displaystyle g({\mathrm{\nabla }}_X(Y_1+Y_2),Z)=g({\mathrm{\nabla }}_X{Y}_1,Z)+g({\mathrm{\nabla }}_X{Y}_2,Z)}$

${\displaystyle g({\mathrm{\nabla }}_X(fY),Z)=X(f)g(Y,Z)+fg({\mathrm{\nabla }}_{X}Y,Z)}$

${\displaystyle g({\mathrm{\nabla }}_{X}Y,Z)+g({\mathrm{\nabla }}_{X}Z,Y)=X(g(Y,Z))}$

${\displaystyle g({\mathrm{\nabla }}_{X}Y,Z)-g({\mathrm{\nabla }}_{Y}X,Z)=g([X,Y],Z).}$

Hence the Koszul expression does, in fact, define a connection, and this connection is compatible with the metric and is torsion free, i.e. is a Levi-Civita connection.

With minor variation, the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.

Let${\displaystyle \mathrm{\nabla }}$ be an affine connection on the tangent bundle. Choose local coordinates${\displaystyle x^1,\dots ,x^n}$ with coordinate basis vector fields${\displaystyle {\mathrm{\partial }}_1,\dots ,{\mathrm{\partial }}_n}$ and write${\displaystyle {\mathrm{\nabla }}_j}$ for${\displaystyle {\mathrm{\nabla }}_{{\mathrm{\partial }}_j}}$. TheChristoffel symbols${\displaystyle {\mathrm{\Gamma }}_{jk}^l}$ of${\displaystyle \mathrm{\nabla }}$ with respect to these coordinates are defined as

${\displaystyle {\mathrm{\nabla }}_j{\mathrm{\partial }}_k={\mathrm{\Gamma }}_{jk}^l{\mathrm{\partial }}_l}$

The Christoffel symbols conversely define the connection${\displaystyle \mathrm{\nabla }}$ on the coordinate neighbourhood because

that is,

${\displaystyle ({\mathrm{\nabla }}_{j}Y{)}^l={\mathrm{\partial }}_j{Y}^l+{\mathrm{\Gamma }}_{jk}^l{Y}^k}$

An affine connection${\displaystyle \mathrm{\nabla }}$ is compatible with a metric iff

${\displaystyle {\mathrm{\partial }}_i(g({\mathrm{\partial }}_j,{\mathrm{\partial }}_k))=g({\mathrm{\nabla }}_i{\mathrm{\partial }}_j,{\mathrm{\partial }}_k)+g({\mathrm{\partial }}_j,{\mathrm{\nabla }}_i{\mathrm{\partial }}_k)=g({\mathrm{\Gamma }}_{ij}^l{\mathrm{\partial }}_l,{\mathrm{\partial }}_k)+g({\mathrm{\partial }}_j,{\mathrm{\Gamma }}_{ik}^l{\mathrm{\partial }}_l)}$

i.e., if and only if

${\displaystyle {\mathrm{\partial }}_i{g}_{jk}={\mathrm{\Gamma }}_{ij}^l{g}_{lk}+{\mathrm{\Gamma }}_{ik}^l{g}_{jl}.}$

An affine connection∇ is torsion free iff

${\displaystyle {\mathrm{\nabla }}_j{\mathrm{\partial }}_k-{\mathrm{\nabla }}_k{\mathrm{\partial }}_j=({\mathrm{\Gamma }}_{jk}^l-{\mathrm{\Gamma }}_{kj}^l){\mathrm{\partial }}_l=[{\mathrm{\partial }}_j,{\mathrm{\partial }}_k]=0.}$

i.e., if and only if

${\displaystyle {\mathrm{\Gamma }}_{jk}^l={\mathrm{\Gamma }}_{kj}^l}$

is symmetric in its lower two indices.

As one checks by taking for${\displaystyle X,Y,Z}$, coordinate vector fields${\displaystyle {\mathrm{\partial }}_j,{\mathrm{\partial }}_k,{\mathrm{\partial }}_l}$ (or computes directly), the Koszul expression of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

${\displaystyle {\mathrm{\Gamma }}_{jk}^l=\frac{1}{2}{g}^{lr}\left({\mathrm{\partial }}_k{g}_{rj}+{\mathrm{\partial }}_j{g}_{rk}-{\mathrm{\partial }}_r{g}_{jk}\right)}$

where as usual${\displaystyle g^{ij}}$ are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix${\displaystyle g_{kl}}$.

The Levi-Civita connection (like any affine connection) also defines a derivative alongcurves, sometimes denoted byD.

Given a smooth curveγ on(M,g) and avector fieldV alongγ its derivative is defined by

${\displaystyle D_{t}V={\mathrm{\nabla }}_{\dot{\gamma }(t)}V.}$

Formally,D is thepullback connectionγ*∇ on thepullback bundleγ*TM.

In particular,${\displaystyle \dot{\gamma }(t)}$ is a vector field along the curveγ itself. If${\displaystyle {\mathrm{\nabla }}_{\dot{\gamma }(t)}\dot{\gamma }(t)}$ vanishes, the curve is called a geodesic of the covariant derivative. Formally, the condition can be restated as the vanishing of the pullback connection applied to${\displaystyle \dot{\gamma }}$:

${\displaystyle \left({\gamma }^{\ast }\mathrm{\nabla }\right)\dot{\gamma }\equiv 0.}$

If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely thosegeodesics of themetric that are parametrised proportionally to their arc length.

In general,parallel transport along a curve with respect to a connection definesisomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms areorthogonal – that is, they preserve the inner products on the various tangent spaces.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on thepunctured plane${\displaystyle {\mathbf{R}}^2\mathrm{\setminus }\{0,0\}}$. The curve the parallel transport is done along is the unit circle. Inpolar coordinates, the metric on the left is the standardEuclidean metric${\displaystyle d{s}^2=d{x}^2+d{y}^2=d{r}^2+r^{2}d{\theta }^2}$, while the metric on the right is${\displaystyle d{s}^2=d{r}^2+d{\theta }^2}$. The first metric extends to the entire plane, but the second metric has a singularity at the origin:

${\displaystyle dr=\frac{xdx+ydy}{\sqrt{x^2+y^2}}}$

${\displaystyle d\theta =\frac{xdy-ydx}{x^2+y^2}}$

${\displaystyle d{r}^2+d{\theta }^2=\frac{(xdx+ydy{)}^2}{x^2+y^2}+\frac{(xdy-ydx{)}^2}{(x^2+y^2{)}^2}}$.

Parallel transports on the punctured plane under Levi-Civita connections

This transport is given by the metric${\displaystyle d{s}^2=d{r}^2+r^{2}d{\theta }^2}$.

This transport is given by the metric${\displaystyle d{s}^2=d{r}^2+d{\theta }^2}$.

Warning: This is parallel transport on the punctured planealong the unit circle, not parallel transporton the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

Let⟨ , ⟩ be the usualscalar product onR³. LetS² be theunit sphere inR³. The tangent space toS² at a pointm is naturally identified with the vector subspace ofR³ consisting of all vectors orthogonal tom. It follows that a vector fieldY onS² can be seen as a mapY :S² →R³, which satisfies$⟨Y(m),m⟩=0,\mathrm{\forall }m\in {\mathbf{S}}^2.$

Denote asd_mY the differential of the mapY at the pointm. Then we have:

In fact, this connection is the Levi-Civita connection for the metric onS² inherited fromR³. Indeed, one can check that this connection preserves the metric.

If the metric${\displaystyle g}$ in aconformal class is replaced by the conformally rescaled metric of the same class${\displaystyle \hat{g}=e^{2\gamma }g}$, then the Levi-Civita connection transforms according to the rule^[12]$${\displaystyle {\hat{\mathrm{\nabla }}}_{X}Y={\mathrm{\nabla }}_{X}Y+X(\gamma )Y+Y(\gamma )X-g(X,Y){\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}_g(\gamma ).}$$where${\displaystyle {\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}_g(\gamma )}$ is the gradient vector field of${\displaystyle \gamma }$ i.e. the vector field${\displaystyle g}$-dual to${\displaystyle d\gamma }$, in local coordinates given by${\displaystyle g^{ik}({\mathrm{\partial }}_i\gamma ){\mathrm{\partial }}_k}$. Indeed, it is trivial to verify that${\displaystyle \hat{\mathrm{\nabla }}}$ is torsion-free. To verify metricity, assume that${\displaystyle g(Y,Y)}$ is constant. In that case,$${\displaystyle \hat{g}({\hat{\mathrm{\nabla }}}_{X}Y,Y)=X(\gamma )\hat{g}(Y,Y)=\frac{1}{2}X(\hat{g}(Y,Y)).}$$

As an application, consider again the unit sphere, but this time understereographic projection, so that the metric (in complexFubini–Study coordinates${\displaystyle z,\overline{z}}$) is:$${\displaystyle g=\frac{4dzd\overline{z}}{(1+z\overline{z}{)}^2}.}$$This exhibits the metric of the sphere as conformally flat, with the Euclidean metric${\displaystyle dzd\overline{z}}$, with${\displaystyle \gamma =\ln (2)-\ln (1+z\overline{z})}$. We have${\displaystyle d\gamma =-(1+z\overline{z}{)}^{-1}(\overline{z}dz+zd\overline{z})}$, and so$${\displaystyle {\hat{\mathrm{\nabla }}}_{{\mathrm{\partial }}_z}{\mathrm{\partial }}_z=-\frac{2\overline{z}{\mathrm{\partial }}_z}{1+z\overline{z}}.}$$With the Euclidean gradient${\displaystyle {\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}_{Euc}(\gamma )=-(1+z\overline{z}{)}^{-1}(\overline{z}{\mathrm{\partial }}_z+z{\mathrm{\partial }}_{\overline{z}})}$, we have$${\displaystyle {\hat{\mathrm{\nabla }}}_{{\mathrm{\partial }}_z}{\mathrm{\partial }}_{\overline{z}}=0.}$$These relations, together with their complex conjugates, define the Christoffel symbols for the two-sphere.

• Weitzenböck connection

• Boothby, William M. (1986).An introduction to differentiable manifolds and Riemannian geometry. Academic Press.ISBN 0-12-116052-1.
• Kobayashi, Shoshichi;Nomizu, Katsumi (1963).Foundations of differential geometry. John Wiley & Sons.ISBN 0-470-49647-9.{{cite book}}:ISBN / Date incompatibility (help) See Volume I pag. 158

• "Levi-Civita connection",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection
• Levi-Civita connection at the Manifold Atlas

• Riemannian geometry
• Connection (mathematics)

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