Recurrent tensor
Recurrent tensors or recurrent tensor fields are used in the mathematical field of differential geometry .
definition
In differential geometry, a recurrent tensor is defined as follows: Let a connection on a manifold M . A tensor A (in the sense of a tensor field ) is called recurrent with respect to the relationship if there is a one-form on M such that
- .
Examples
Parallel tensors
Examples of recurrent tensors are parallel tensors with regard to a relationship ( ).
An example of a parallel tensor is a (semi-) Riemannian metric with regard to its Levi-Civita relationship . So let M be a manifold with metric g , then the Levi-Civita connection is defined by the metric and it follows from the definition
- .
Another example are recurrent vector fields , where in special cases parallel vector fields can be derived from recurrent vector fields. For this, let a semiriemannian manifold and a recurent vector field with
- ,
so it follows from ( closed) that it can be rescaled to a parallel vector field. In particular, each vector field with a non-vanishing length can be rescaled into a parallel vector field. Non-parallel recurrent vector fields are therefore in particular light-like .
Metric space
Another example of a recurrent tensor appears in connection with Weyl structures . Historically, the Weyl structure emerged from Hermann Weyl's considerations on the properties of parallel displacement of vectors and their length. The requirement to be able to describe a variety locally affine, a condition is created to the connected with the affine parallel shift relation . It must be torsion-free :
- .
For the additional parallel shift of the metric, he demanded as a special property that, although not the length, but the length ratio of parallel shifted vector fields should be retained. The relationship defined in this way then fulfills the property
for a one-shape . In particular, the metric is therefore a recurrent tensor with respect to . Weyl now called the resulting manifold with affine connection and recurrent metric g metric space. Strictly speaking, Weyl was not only looking at a metric, but the conformal structure over g . This can be motivated as follows:
Under a conformal change , the form transforms , thereby inducing a mapping onto the manifold with conformal structure . To do this, you fix and and define:
- .
thus fulfills the conditions of a Weyl structure:
- .
literature
- Hermann Weyl : Meeting reports of the Prussian Academy of Sciences in Berlin . 1918, p. 465-480 .
- AG Walker: On parallel fields of partially null vector spaces. (PDF) In: The Quarterly Journal of Mathematics. Ser. 1, 20, 1949, pp. 135-145, ISSN 0033-5606 .
- EM Patterson: On symmetric recurrent tensors of the second order. (PDF) In: The Quarterly Journal of Mathematics. 2, 1, 1950, pp. 151-158.
- Yung-Chow Wong: Recurrent Tensors on a Linearly Connected Differentiable Manifold. In: Transactions of the American Mathematical Society. 99, 2, May 1961, pp. 325-341, ISSN 0002-9947 .
- Gerald B. Folland: Weyl Manifolds. (PDF; 113 kB). In: Journal of Differential Geometry. 4, 1970, pp. 145-153, ISSN 0022-040X .
- Dmitri V. Alekseevsky, Helga Baum (Ed.): Recent developments in pseudo-Riemannian geometry . European Mathematical Society, Zurich 2008, ISBN 978-3-03719-051-7 .