Levi-Civita connection

An affine connection on a Riemannian space $ M $ 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 $ M $ is determined uniquely by these conditions, hence every Riemannian space $ M $ 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 $ M $, where $ d s ^ {2} = g _ {ij} d x ^ {i} d x ^ {j} $, the Levi-Civita connection on $ M $ is defined by the forms $ \omega _ {j} ^ {i} = \{ _ {jk} ^ { i } \} d x ^ {k} $, where

$$ \left \{ \begin{array}{c} i \\ jk \end{array} \right \} = \frac{1}{2} g ^ {il} \left ( \frac{\partial g _ {lj} }{\partial x ^ {k} } + \frac{\partial g _ {lk} }{\partial x ^ {j} } - \frac{\partial g _ {jk} }{\partial x ^ {l} } \right ) ; $$

its curvature tensor $ R _ {jkl} ^ {i} $ is defined by the formula

$$ d \omega _ {j} ^ {i} + \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} = \frac{1}{2} R _ {jkl} ^ {i} d x ^ {k} \wedge d x ^ {l} . $$

Let $ R _ {ij,kl} = g _ {im} R _ {jkl} ^ {m} $; then

$$ R _ {ij,kl} = \frac{1}{2} \left \{ \frac{\partial ^ {2} g _ {jk} }{\partial x ^ {i} \partial x ^ {l} } - \frac{\partial ^ {2} g _ {jl} }{\partial x ^ {i} \partial x ^ {k} } - \frac{\partial ^ {2} g _ {ik} }{\partial x ^ {j} \partial x ^ {l} } + \frac{\partial ^ {2} g _ {il} }{\partial x ^ {j} \partial x ^ {k} } \right \} + $$

$$ + g _ {pq} \left ( \left \{ \begin{array}{c} p \\ il \end{array} \right \} \left \{ \begin{array}{c} q \\ jk \end{array} \right \} - \left \{ \begin{array}{c} p \\ ik \end{array} \right \} \left \{ \begin{array}{c} q \\ jl \end{array} \right \} \right ) ; $$

thus:

$$ R _ {ij,kl} = - R _ {ij,lk} ,\ \ R _ {ij,kl} = R _ {kl,ij} , $$

$$ R _ {ij,kl} + R _ {ik,lj} + R _ {il,jk} = 0 . $$

The curvature tensor of the Levi-Civita connection has $ n ^ {2} ( n ^ {2} - 1 ) / 12 $ essential components, where $ n = \mathop{\rm dim} M $. For example, for $ n = 2 $ there is only one essential component: $ R _ {12,12} = K \mathop{\rm det} | g _ {ij} | $, where $ K $ is the Gaussian curvature.

If a Riemannian space $ M $ is isometrically immersed in a Euclidean space $ E ^ {N} $, then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields $ X $, $ Y $ on $ M \subset E ^ {N} $ the covariant derivative $ ( \nabla _ {Y} X ) _ {x} $ at a point $ x \in M $ is the orthogonal projection on the tangent plane $ T _ {x} ( M) \subset E ^ {N} $ of the ordinary differential $ ( d _ {Y} X ) _ {x} $ of the field $ X $ in $ E ^ {N} $ with respect to the vector $ Y _ {x} \in T _ {x} ( M) $. 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)