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 ${\displaystyle \nabla }$  ${\displaystyle \nabla }$ ${\displaystyle \omega }$

${\displaystyle \nabla A=\omega \otimes A}$.

Examples

Parallel tensors

Examples of recurrent tensors are parallel tensors with regard to a relationship ( ). ${\displaystyle \nabla A=0}$

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 ${\displaystyle \nabla ^{LC}}$

${\displaystyle \nabla ^{LC}g=0}$.

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 ${\displaystyle (M,g)}$ ${\displaystyle X}$

${\displaystyle \nabla X=\omega \otimes X}$,

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 . ${\displaystyle d\omega =0}$ ${\displaystyle \omega }$ ${\displaystyle X}$

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 : ${\displaystyle \nabla }$

${\displaystyle T^{\nabla }(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]=0}$.

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 ${\displaystyle \nabla '}$

${\displaystyle \nabla 'g=\varphi \otimes g}$

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: ${\displaystyle \varphi }$ ${\displaystyle \nabla '}$ ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$ ${\displaystyle [g]}$

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: ${\displaystyle g\rightarrow e^{\lambda }g}$ ${\displaystyle \phi }$ ${\displaystyle \varphi \rightarrow \varphi -d\lambda }$ ${\displaystyle F:[g]\rightarrow \Lambda ^{1}(M)}$ ${\displaystyle (M,[g])}$ ${\displaystyle [g]}$ ${\displaystyle g}$ ${\displaystyle \varphi }$

${\displaystyle F(e^{\lambda }g):=\varphi -d\lambda }$.

${\displaystyle F}$ thus fulfills the conditions of a Weyl structure:

${\displaystyle F(e^{\lambda }g)=F(g)-d\lambda }$.

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 .

Individual evidence

1. ↑ Alekseevsky, Baum (2008)
2. ↑ Weyl (1918)
3. ^ Folland (1970)