Parallelism, absolute
A (global) field of frames on a manifold (cf. Frame). An absolute parallelism determines an isomorphism of all tangent spaces of the manifold
, under which the tangent vectors of the spaces
and
having the same coordinates with respect to the frames
and
are identified. This assigns to the manifold a linear connection
of zero curvature. The parallel fields relative to this connection are the tensor fields having constant coordinates with respect to the field of frames
(in particular, the vector fields
are parallel), and the operation of covariant differentiation of a tensor field
in the direction of a vector field
reduces to differentiation of the coordinates of
relative to
in the direction of
. Conversely, a linear connection
with zero curvature on a simply-connected manifold
determines an absolute parallelism
if there is given in addition a frame
in some tangent space
. The corresponding absolute parallelism
is obtained from the frame
by extension using parallel displacement of the connection
(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 -structures (cf.
-structure), an absolute parallelism is a
-structure, where
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
for which
,
. For this it is necessary and sufficient that the vector fields
commute in pairs, in other words, that the torsion tensor
of the connection
, defined by the formula
, 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
is geodesically complete. In the integrable case, the completeness of the vector fields
is sufficient for this. A complete integrable absolute parallelism on a simply-connected manifold
determines on
the structure of an affine space. More generally, a complete absolute parallelism with a covariantly constant torsion tensor
on a simply-connected manifold
with a distinguished point determines on
the structure of a Lie group, with structure constants
, for which the fields
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 . 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) |
Parallelism, absolute. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallelism,_absolute&oldid=33873