# 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

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.^{[6]} In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results.^{[7]} In the same year, Hermann Weyl generalized
Levi-Civita's results.^{[8]}^{[9]}

## Notation

- (
*M*,*g*) denotes a Riemannian or pseudo-Riemannian manifold. *TM*is the tangent bundle of*M*.*g*is the Riemannian or pseudo-Riemannian metric of*M*.*X*,*Y*,*Z*are smooth vector fields on*M*, i. e. smooth sections of*TM*.- [
*X*,*Y*] is the Lie bracket of*X*and*Y*. It is again a smooth vector field.

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 *X*_{p}, *Y*_{p} 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

where Einstein's summation convention is used.

## Formal definition

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

*it preserves the metric*, i.e., ∇*g*= 0.*it is torsion-free*, i.e., for any vector fields*X*and*Y*we have ∇_{X}*Y*− ∇_{Y}*X*= [*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:

By condition 2, the right hand side is equal to

so we find the Koszul formula

Since *Z* is arbitrary, this uniquely determines ∇_{X}*Y*. 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 *x*^{1} … *x ^{n}* and let Γ

*be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry*

^{l}_{jk}The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as

where as usual *g ^{ij}* are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix (

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

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*:

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 , while the metric on the right has standard form in polar coordinates, and thus preserves the vector tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates:

## Example: the unit sphere in **R**^{3}

Let ⟨ , ⟩ be the usual scalar product on **R**^{3}. Let **S**^{2} be the unit sphere in **R**^{3}. The tangent space to **S**^{2} at a point *m* is naturally identified with the vector subspace of **R**^{3} consisting of all vectors orthogonal to *m*. It follows that a vector field *Y* on **S**^{2} can be seen as a map *Y* : **S**^{2} → **R**^{3}, which satisfies

Denote as *d _{m}Y*(

*X*) the covariant derivative of the map

*Y*in the direction of the vector

*X*. Then we have:

**Lemma:**The formula- defines an affine connection on
**S**^{2}with vanishing torsion.

**Proof:**It is straightforward to prove that ∇ satisfies the Leibniz identity and is*C*^{∞}(**S**^{2}) 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**S**^{2}- Consider the map
*f*that sends every*m*in**S**^{2}to ⟨*Y*(*m*),*m*⟩, which is always 0. The map*f*is constant, hence its differential vanishes. In particular - The equation (1) above follows. Q.E.D.

In fact, this connection is the Levi-Civita connection for the metric on **S**^{2} inherited from **R**^{3}. Indeed, one can check that this connection preserves the metric.

## See also

## Notes

- ↑ See Levi-Civita (1917)
- ↑ See Christoffel (1869)
- ↑ See Spivak (1999) Volume II, page 238
- ↑
Brouwer, L. E. J. (1906), "Het krachtveld der niet-Euclidische, negatief gekromde ruimten",
*Koninklijke Akademie van Wetenschappen. Verslagen*,**15**: 75–94 - ↑
Brouwer, L. E. J. (1906), "The force field of the non-Euclidean spaces with negative curvature",
*Koninklijke Akademie van Wetenschappen. Proceedings*,**9**: 116–133 - ↑
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, (Subscription required (help)) - ↑
Schouten, Jan Arnoldus (1918), "Die direkte Analysis zur neueren Relativiteitstheorie",
*Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam*,**12**(6): 95 - ↑
Weyl, Hermann (1918), "Gravitation und Elektrizitat",
*Sitzungsberichte Berliner Akademie*: 465–480 - ↑
Weyl, Hermann (1918), "Reine Infinitesimal geometrie",
*Mathematische Zeitschrift*,**2**: 384–411, doi:10.1007/bf01199420

## References

### Primary historical references

- Christoffel, Elwin B. (1869), "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades",
*Journal für die reine und angewandte Mathematik*,**70**: 46–70, doi:10.1515/crll.1869.70.46 - Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura Riemanniana",
*Rend. Circ. Mat. Palermo*,**42**: 73–205, doi:10.1007/bf03014898

### Secondary references

- 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. See Volume I pag. 158 - Spivak, Michael (1999).
*A Comprehensive introduction to differential geometry (Volume II)*. Publish or Perish Press. ISBN 0-914098-71-3.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Levi-Civita connection",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - MathWorld: Levi-Civita Connection
- PlanetMath: Levi-Civita Connection
- Levi-Civita connection at the Manifold Atlas