View source for Parallelism, absolute

A (global) field of frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714001.png" /> on a [[Manifold|manifold]] (cf. [[Frame|Frame]]). An absolute parallelism determines an isomorphism of all tangent spaces of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714002.png" />, under which the tangent vectors of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714004.png" /> having the same coordinates with respect to the frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714006.png" /> are identified. This assigns to the manifold a [[Linear connection|linear connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714007.png" /> of zero curvature. The parallel fields relative to this connection are the tensor fields having constant coordinates with respect to the field of frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714008.png" /> (in particular, the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p0714009.png" /> are parallel), and the operation of [[Covariant differentiation|covariant differentiation]] of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140010.png" /> in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140011.png" /> reduces to differentiation of the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140012.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140013.png" /> in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140014.png" />. Conversely, a linear connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140015.png" /> with zero curvature on a simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140016.png" /> determines an absolute parallelism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140017.png" /> if there is given in addition a frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140018.png" /> in some tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140019.png" />. The corresponding absolute parallelism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140020.png" /> is obtained from the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140021.png" /> by extension using [[Parallel displacement(2)|parallel displacement]] of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140022.png" /> (the parallel displacement does not depend on the choice of the path connecting two given points of the manifold if the connection has zero curvature and the manifold is simply connected).

From the point of view of the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140023.png" />-structures (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140024.png" />-structure]]), an absolute parallelism is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140025.png" />-structure, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140026.png" /> is the group consisting of one (identity) element. The integrability of such a structure means that in a neighbourhood of any point of the manifold there exists a system of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140027.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140029.png" />. For this it is necessary and sufficient that the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140030.png" /> commute in pairs, in other words, that the [[Torsion tensor|torsion tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140031.png" /> of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140032.png" />, defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140033.png" />, is identically equal to zero. An absolute parallelism is called complete if all vector fields having constant coordinates with respect to the field of frames are complete, or equivalently, if the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140034.png" /> is geodesically complete. In the integrable case, the completeness of the vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140035.png" /> is sufficient for this. A complete integrable absolute parallelism on a simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140036.png" /> determines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140037.png" /> the structure of an affine space. More generally, a complete absolute parallelism with a covariantly constant torsion tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140038.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140039.png" /> on a simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140040.png" /> with a distinguished point determines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140041.png" /> the structure of a Lie group, with structure constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140042.png" />, for which the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140043.png" /> form a basis for the space of left-invariant fields.

The group of automorphisms of an absolute parallelism is a Lie group which acts freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071400/p07140044.png" />. Necessary and sufficient conditions for two absolute parallelisms to be locally isomorphic are known (see [[#References|[3]]]). They are expressed in terms of the torsion tensor and its covariant derivative.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Nomizu,   "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>