Levi-Civita connection

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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.


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.




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.




where Einstein's summation convention is used.

Formal definition[edit]

An affine connection is called a Levi-Civita connection if


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

Assuming 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

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[edit]

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



Derivative along curve[edit]



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



Parallel transport[edit]


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 right metric has standard form in polar coordinates, and thus preserves the vector tangent to the circle. This metric has a singularity at the origin in Euclidean coordinates.

Parallel Transports Under Levi-Civita Connections
Cartesian transport
This transport is given by the metric ds2 = dr2 + r2dθ2.
Polar transport
This transport is given by the metric ds2 = dr2 + dθ2.

Example: The unit sphere in R3[edit]

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 sub-space 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


Lemma: The formula

defines an affine connection on S2 with vanishing torsion.


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

The equation (1) above follows.


See also[edit]


  1. ^ See Levi-Civita (1917)
  2. ^ See Christoffel (1869)
  3. ^ See Spivak (1999) Volume II, page 238


Primary historical references[edit]

  • Christoffel, Elwin Bruno (1869), "Über die Transformation der homogenen Differentialausdrücke zweiten Grades", J. für die Reine und Angew. Math. 70: 46–70 
  • Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana", Rend. Circ. Mat. Palermo 42: 73–205, doi:10.1007/bf03014898 

Secondary references[edit]

  • Boothby, William M. (1986). An introduction to differentiable manifolds and Riemannian geometry. Academic Press. ISBN 0-12-116052-1. 
  • Kobayashi, S.; Nomizu, K. (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[edit]