Levi-Civita connection

An affine connection on a Riemannian space that is a Riemannian connection (that is, a connection with respect to which the metric tensor is covariantly constant) and has zero torsion. An affine connection on is determined uniquely by these conditions, hence every Riemannian space has a unique Levi-Civita connection. This concept first arose in 1917 with T. Levi-Civita [1] as the concept of parallel displacement of a vector in Riemannian geometry. The idea itself goes back to F. Minding, who in 1837 introduced the concept of the involute of a curve on a surface.

With respect to a local coordinate system in , where , the Levi-Civita connection on is defined by the forms , where

its curvature tensor is defined by the formula

Let ; then

thus:

The curvature tensor of the Levi-Civita connection has essential components, where . For example, for there is only one essential component: , where is the Gaussian curvature.

If a Riemannian space is isometrically immersed in a Euclidean space , then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields , on the covariant derivative at a point is the orthogonal projection on the tangent plane of the ordinary differential of the field in with respect to the vector . In other words, the mapping of a neighbouring infinitely close tangent plane onto the original tangent plane is accomplished by orthogonal projection.

References

[1]	T. Levi-Civita, "Nozione di parallelismo in una varietá qualunque e consequente specificazione geometrica della curvatura riemanniana" Rend. Circ. Math. Palermo , 42 (1917) pp. 173–205
[2]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[3]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)