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 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. 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 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
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 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 local coordinates , the action reads
Hence, if a Levi-Civita connection exists, it must be unique, because is arbitrary, is non degenerate, and the right hand side does not depend on .
To prove existence, note that for given vector field and , the right hand side of the Koszul expression is function-linear in the vector field , not just real linear. Hence by the non degeneracy of , the right hand side uniquely defines some new vector field which we suggestively denote as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields , and all functions
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.
Let be an affine connection on the tangent bundle. Choose local coordinates with coordinate basis vector fields and write for . The Christoffel symbols of with respect to these coordinates are defined as
The Christoffel symbols conversely define the connection on the coordinate neighbourhood because
An affine connection is compatible with a metric iff
i.e., if and only if
An affine connection ∇ is torsion free iff
i.e., if and only if
is symmetric in its lower two indices.
As one checks by taking for , coordinate vector fields (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
where as usual are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix .
Derivative along curveEdit
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 fieldV along γ its derivative is defined by
In particular, is a vector field along the curve γ itself. If 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 :
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.
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 (when ), 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:
Parallel transports under Levi-Civita connections
This transport is given by the metric .
This transport is given by the metric .
Example: the unit sphere in R3Edit
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
Denote as dmY(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 S2 with vanishing torsion.
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
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
^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. S2CID 122999847.
^See Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume II). Publish or Perish Press. p. 238. ISBN 0-914098-71-3.
^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. Bibcode:1906KNAB....9..116B.
^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.