The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subject to the condition of vanishing torsion, gives a description of Einstein gravity. For supersymmetry, the same constraint, of vanishing torsion, gives (the field equations of) 11-dimensional supergravity. That is, the contorsion tensor, along with the connection, becomes one of the dynamical objects of the theory, demoting the metric to a secondary, derived role.
The elimination of torsion in a connection is referred to as the absorption of torsion, and is one of the steps of Cartan's equivalence method for establishing the equivalence of geometric structures.
In metric geometry, the contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol and the unique torsion-free Levi-Civita connection for the same metric.
The contorsion tensor is defined in terms of the torsion tensor as (up to a sign, see below)
where the indices are being raised and lowered with respect to the metric:
The reason for the non-obvious sum in the definition of the contorsion tensor is due to the sum-sum difference that enforces metric compatibility. The contorsion tensor is antisymmetric in the first two indices, whilst the torsion tensor itself is antisymmetric in its last two indices; this is shown below.
The full metric compatible affine connection can be written as:
Where the torsion-free Levi-Civita connection:
In affine geometry, one does not have a metric nor a metric connection, and so one is not free to raise and lower indices on demand. One can still achieve a similar effect by making use of the solder form, allowing the bundle to be related to what is happening on its base space. This is an explicitly geometric viewpoint, with tensors now being geometric objects in the vertical and horizontal bundles of a fiber bundle, instead of being indexed algebraic objects defined only on the base space.
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En géométrie différentielle, la torsion constitue, avec la courbure, une mesure de la façon dont une base mobile évolue le long des courbes, et le tenseur de torsion en donne l'expression générale dans le cadre des variétés, c'est-à-dire des « espaces courbes » de toutes dimensions. La torsion se manifeste en géométrie différentielle classique comme une valeur numérique associée à chaque point d'une courbe de l'espace euclidien.
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to: A connection for which the covariant derivatives of the metric on E vanish. A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is a Riemannian connection; there is a unique such which is torsion free, the Levi-Civita connection.
En géométrie différentielle, une connexion d'Ehresmann (d'après le mathématicien français Charles Ehresmann qui a le premier formalisé ce concept) est une version de la notion de connexion qui est définie sur des fibrés. En particulier, elle peut être non-linéaire, puisqu'un espace fibré n'a pas de notion de linéarité qui lui soit naturellement adaptée. Cependant, une connexion de Koszul (parfois aussi appelée connexion linéaire) en est un cas particulier.
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