# 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)