# Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

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 as differential operators on smooth functions. In a basis, the action reads

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

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.

If a Levi-Civita connection exists, it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor g we 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 equal to

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

so we find the Koszul formula

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

Since Z is arbitrary, this uniquely determines XY. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi-Civita connection.

## Christoffel symbols

Let be the connection of the Riemannian metric. Choose local coordinates x1xn and let Γljk be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry

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

The definition 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 gij are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix (gkl).

## 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 γ on (M ,g) and a vector field V along γ its derivative is defined by

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

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

In particular, γ̇(t) is a vector field along the curve γ itself. If γ̇(t)γ̇(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 γ̇:

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

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 : S2R3, 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.