View source for Levi-Civita connection

An [[Affine connection|affine connection]] on a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582301.png" /> that is a [[Riemannian connection|Riemannian connection]] (that is, a connection with respect to which the [[Metric tensor|metric tensor]] is covariantly constant) and has zero [[Torsion|torsion]]. An affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582302.png" /> is determined uniquely by these conditions, hence every Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582303.png" /> has a unique Levi-Civita connection. This concept first arose in 1917 with T. Levi-Civita [[#References|[1]]] as the concept of [[Parallel displacement(2)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582305.png" />, the Levi-Civita connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582306.png" /> is defined by the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582307.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582308.png" /></td> </tr></table>

its [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582309.png" /> is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823010.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823011.png" />; then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823012.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823013.png" /></td> </tr></table>

thus:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823014.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823015.png" /></td> </tr></table>

The curvature tensor of the Levi-Civita connection has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823016.png" /> essential components, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823017.png" />. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823018.png" /> there is only one essential component: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823020.png" /> is the [[Gaussian curvature|Gaussian curvature]].

If a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823021.png" /> is isometrically immersed in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823022.png" />, then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823025.png" /> the [[Covariant derivative|covariant derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823026.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823027.png" /> is the orthogonal projection on the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823028.png" /> of the ordinary differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823029.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823031.png" /> with respect to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823032.png" />. In other words, the mapping of a neighbouring infinitely close tangent plane onto the original tangent plane is accomplished by orthogonal projection.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Gromoll,   W. Klingenberg,   W. Meyer,   "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>