View source for Christoffel symbol

''of a differential quadratic form

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''

An abbreviated notation for the expression

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The symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222203.png" /> are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222204.png" />, defined by

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222206.png" /> is defined as follows:

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These symbols were introduced by E.B. Christoffel in 1869.

====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222209.png" />, be a [[Linear connection|linear connection]] on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222011.png" /> denotes the space of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222013.png" /> be a chart of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222014.png" />. Then on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222016.png" /> is completely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222018.png" /> are coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222019.png" />. The Christoffel symbols of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222020.png" /> are now given by

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It is important to note that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222022.png" /> are not the components of a tensor field. In fact if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222023.png" /> denote the Christoffel symbols of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222024.png" /> with respect to a second set of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222026.png" />, then

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Let now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222028.png" /> be the Riemannian connection (cf. [[Riemannian geometry|Riemannian geometry]]) defined by a (local) Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222029.png" />. Then the Christoffel symbols of this quadratic differential form are those of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222030.png" />. I.e.,

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so that indeed

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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222033.png" /> are the Christoffel symbols of the second kind of the quadratic differential form as defined above.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.S. Millman,   G.D. Parker,   "Elements of differential geometry" , Prentice-Hall  (1977)  pp. Chapt. 7</TD></TR></table>