Parallelism, absolute

A (global) field of frames $e=(e_1,\dots,e_n)$ on a manifold (cf. Frame). An absolute parallelism determines an isomorphism of all tangent spaces of the manifold $M$, under which the tangent vectors of the spaces $T_pM$ and $T_qM$ having the same coordinates with respect to the frames $e_p$ and $e_q$ are identified. This assigns to the manifold a linear connection $\nabla^e$ of zero curvature. The parallel fields relative to this connection are the tensor fields having constant coordinates with respect to the field of frames $e$ (in particular, the vector fields $e_1,\dots,e_n$ are parallel), and the operation of covariant differentiation of a tensor field $T$ in the direction of a vector field $X$ reduces to differentiation of the coordinates of $T$ relative to $e$ in the direction of $X$. Conversely, a linear connection $\nabla$ with zero curvature on a simply-connected manifold $M$ determines an absolute parallelism $e$ if there is given in addition a frame $e_p$ in some tangent space $T_pM$. The corresponding absolute parallelism $e$ is obtained from the frame $e_p$ by extension using parallel displacement of the connection $\nabla$ (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 $G$-structures (cf. $G$-structure), an absolute parallelism is a $\{1\}$-structure, where $\{1\}$ 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 $x^i$ for which $e_i=\partial/\partial x^i$, $i=1,\dots,n$. For this it is necessary and sufficient that the vector fields $e_1,\dots,e_n$ commute in pairs, in other words, that the torsion tensor $C=C_{jk}^i$ of the connection $\nabla^e$, defined by the formula $[e_j,e_k]=C_{jk}^ie_i$, 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 $\nabla^e$ is geodesically complete. In the integrable case, the completeness of the vector fields $e_1,\dots,e_n$ is sufficient for this. A complete integrable absolute parallelism on a simply-connected manifold $M$ determines on $M$ the structure of an affine space. More generally, a complete absolute parallelism with a covariantly constant torsion tensor $C$ $(C_{jk}^i=\text{constant})$ on a simply-connected manifold $M$ with a distinguished point determines on $M$ the structure of a Lie group, with structure constants $C_{jk}^i$, for which the fields $e_i$ 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 $M$. Necessary and sufficient conditions for two absolute parallelisms to be locally isomorphic are known (see [3]). They are expressed in terms of the torsion tensor and its covariant derivative.

References

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