Levi-Civita Connection
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Levi-Civita Connection

In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free.

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.

## History

The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita,[1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols[2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.[3]

The Levi-Civita notions of intrinsic 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

${\displaystyle M^{n}\subset \mathbf {R} ^{\frac {n(n+1)}{2}},}$

since the definition of the Christoffel symbols make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space.

### Remark

In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature.[4][5] In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space.[1] 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]

## Notation

The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of Xp, Yp is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act (by definition) as differential operators on smooth functions. In a local coordinates ${\displaystyle (x_{1},\ldots ,x_{n})}$, the action reads

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

where Einstein's summation convention is used.

## Formal definition

An affine connection ? is called a Levi-Civita connection if

1. it preserves the metric, i.e., ?g = 0.
2. it is torsion-free, i.e., for any vector fields X and Y we have ?XY - ?YX = [X, Y], where [X, Y] is the Lie bracket of the vector fields X and Y.

Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text.

## Fundamental theorem of (pseudo) Riemannian Geometry

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

proof: If a Levi-Civita connection exists, it must be unique. To see this, unravel the definition of the action of a connection on tensors to find

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

Hence we can write condition 1 as

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

By the symmetry of the metric tensor ${\displaystyle g}$ we then find:

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

By condition 2, the right hand side is therefore equal to

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

and we find the Koszul formula

${\displaystyle g(\nabla _{X}Y,Z)={\tfrac {1}{2}}{\Big \{}X{\bigl (}g(Y,Z){\bigr )}+Y{\bigl (}g(Z,X){\bigr )}-Z{\bigl (}g(X,Y){\bigr )}+g([X,Y],Z)-g([Y,Z],X)-g([X,Z],Y){\Big \}}.}$

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 ${\displaystyle \nabla }$.

To prove existence, note that for given vector field ${\displaystyle X}$ and ${\displaystyle Y}$, the right hand side of the Koszul expression is function-linear 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 we suggestively denote ${\displaystyle \nabla _{X}Y}$ 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 ${\displaystyle f}$

${\displaystyle g(\nabla _{X}(Y_{1}+Y_{2}),Z)=g(\nabla _{X}Y_{1},Z)+g(\nabla _{X}Y_{2},Z)}$
${\displaystyle g(\nabla _{X}(fY),Z)=X(f)g(Y,Z)+fg(\nabla _{X}Y,Z)}$
${\displaystyle g(\nabla _{X}Y,Z)+g(\nabla _{X}Z,Y)=X{\bigl (}g(Y,Z){\bigr )}}$
${\displaystyle g(\nabla _{X}Y,Z)-g(\nabla _{Y}X,Z)=g([X,Y],Z).}$

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 (hence the) Levi-Civita connection.

Note that with minor variations the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.

## Christoffel symbols

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

${\displaystyle \nabla _{j}\partial _{k}=\Gamma _{jk}^{l}\partial _{l}}$

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

${\displaystyle \nabla _{X}Y=\nabla _{X^{j}\partial _{j}}Y=X^{j}\nabla _{j}Y=X^{j}\nabla _{j}(Y^{k}\partial _{k})=X^{j}{\bigl (}\partial _{j}(Y^{k})\partial _{k}+Y^{k}\nabla _{j}\partial _{k}{\bigr )}=X^{j}{\bigl (}\partial _{j}(Y^{k})\partial _{k}+Y^{k}\Gamma _{jk}^{l}\partial _{l}{\bigr )}=X^{j}{\bigl (}\partial _{j}(Y^{l})+Y^{k}\Gamma _{jk}^{l}{\bigr )}\partial _{l}}$

ie.

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

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

${\displaystyle \partial _{i}{\bigl (}g(\partial _{j},\partial _{k}){\bigr )}=g(\nabla _{i}\partial _{j},\partial _{k})+g(\partial _{j},\nabla _{i}\partial _{k})=g(\Gamma _{ij}^{l}\partial _{l},\partial _{k})+g(\partial _{j},\Gamma _{ik}^{l}\partial _{l})}$

i.e. iff

${\displaystyle \partial _{i}g_{jk}=\Gamma _{ij}^{l}g_{lk}+\Gamma _{ik}^{l}g_{jl}.}$

An affine connection ? is torsion free iff

${\displaystyle \nabla _{i}\partial _{j}-\nabla _{j}\partial _{i}=(\Gamma _{jk}^{l}-\Gamma _{kj}^{l})\partial _{l}=[\partial _{i},\partial _{j}]=0.}$

i.e. iff

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

is symmetric in its lower two indices.

As one checks by taking for ${\displaystyle X,Y,Z}$, coordinate vectorfields ${\displaystyle \partial _{j},\partial _{k},\partial _{l}}$ (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

${\displaystyle \Gamma _{jk}^{l}={\tfrac {1}{2}}g^{lr}\left(\partial _{k}g_{rj}+\partial _{j}g_{rk}-\partial _{r}g_{jk}\right)}$

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

## Derivative along curve

The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D.

Given a smooth curve ? on (M, g) and a vector field V along ? its derivative is defined by

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

Formally, D is the pullback connection ?*? on the pullback bundle ?*TM.

In particular, ${\displaystyle {\dot {\gamma }}(t)}$ is a vector field along the curve ? itself. If ${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)}$ 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 ${\displaystyle {\dot {\gamma }}}$:

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

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

## Parallel transport

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

The images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in polar coordinates. The metric of left image corresponds to the standard Euclidean metric ${\displaystyle ds^{2}=dx^{2}+dy^{2}=dr^{2}+r^{2}d\theta ^{2}}$, while the metric on the right has standard form in polar coordinates, and thus preserves the vector ${\displaystyle {\partial \over \partial \theta }}$ tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates:

${\displaystyle dr={\frac {xdx+ydy}{\sqrt {x^{2}+y^{2}}}}}$
${\displaystyle d\theta ={\frac {xdy-ydx}{x^{2}+y^{2}}}}$
${\displaystyle dr^{2}+d\theta ^{2}={\frac {(xdx+ydy)^{2}}{x^{2}+y^{2}}}+{\frac {(xdy-ydx)^{2}}{(x^{2}+y^{2})^{2}}}}$
Parallel transports under Levi-Civita connections
This transport is given by the metric ${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}}$.
This transport is given by the metric ${\displaystyle ds^{2}=dr^{2}+d\theta ^{2}}$.

## Example: the unit sphere in R3

Let ? , ? be the usual scalar product on R3. Let S2 be the unit sphere in R3. The tangent space to S2 at a point m is naturally identified with the vector subspace of R3 consisting of all vectors orthogonal to m. It follows that a vector field Y on S2 can be seen as a map Y : S2 -> R3, which satisfies

${\displaystyle {\bigl \langle }Y(m),m{\bigr \rangle }=0,\qquad \forall m\in \mathbf {S} ^{2}.}$

Denote as dmY(X) the covariant derivative of the map Y in the direction of the vector X. Then we have:

Lemma: The formula
${\displaystyle \left(\nabla _{X}Y\right)(m)=d_{m}Y(X)+\langle X(m),Y(m)\rangle m}$
defines an affine connection on S2 with vanishing torsion.
Proof: It is straightforward to prove that ? satisfies the Leibniz identity and is C?(S2) linear in the first variable. It is also a straightforward computation to show that this connection is torsion free. So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all m in S2
${\displaystyle {\bigl \langle }\left(\nabla _{X}Y\right)(m),m{\bigr \rangle }=0\qquad (1).}$
Consider the map f that sends every m in S2 to ?Y(m), m?, which is always 0. The map f is constant, hence its differential vanishes. In particular
${\displaystyle d_{m}f(X)={\bigl \langle }d_{m}Y(X),m{\bigr \rangle }+{\bigl \langle }Y(m),X(m){\bigr \rangle }=0.}$
The equation (1) above follows. Q.E.D.

In fact, this connection is the Levi-Civita connection for the metric on S2 inherited from R3. Indeed, one can check that this connection preserves the metric.

## Notes

1. ^ a b Levi-Civita, Tullio (1917). "Nozione di parallelismo in una varietà qualunque" [The notion of parallelism on any manifold]. Rendiconti del Circolo Matematico di Palermo (in Italian). 42: 173-205. doi:10.1007/BF03014898. JFM 46.1125.02.
2. ^ Christoffel, Elwin B. (1869). "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades". Journal für die reine und angewandte Mathematik. 1869 (70): 46-70. doi:10.1515/crll.1869.70.46.
3. ^ See Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press. p. 238. ISBN 0-914098-71-3.
4. ^ Brouwer, L. E. J. (1906). "Het krachtveld der niet-Euclidische, negatief gekromde ruimten". Koninklijke Akademie van Wetenschappen. Verslagen. 15: 75-94.
5. ^ Brouwer, L. E. J. (1906). "The force field of the non-Euclidean spaces with negative curvature". Koninklijke Akademie van Wetenschappen. Proceedings. 9: 116-133.
6. ^ Schouten, Jan Arnoldus (1918). "Die direkte Analysis zur neueren Relativiteitstheorie". Verhandelingen der Koninklijke Akademie van Wetenschappen Te Amsterdam. 12 (6): 95.
7. ^ Weyl, Hermann (1918). "Gravitation und Elektrizitat". Sitzungsberichte Berliner Akademie: 465-480.
8. ^ Weyl, Hermann (1918). "Reine Infinitesimal geometrie". Mathematische Zeitschrift. 2 (3-4): 384-411. doi:10.1007/bf01199420.