Kulkarni–Nomizu product

In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor.

Definition

If h and k are symmetric (0,2)-tensors, then the product is defined via[1]:

where the Xj are tangent vectors and is the matrix determinant. Note that , as it is clear from the second expression.

With respect to a basis of the tangent space, it takes the compact form

where denotes the total antisymmetrisation symbol.

The Kulkarni–Nomizu product is a special case of the product in the graded algebra

where, on simple elements,

( denotes the symmetric product).

Properties

The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor[2]. For instance, on a 2-dimensional smooth Riemannian manifold, the Riemann curvature tensor has a simple expression in terms of the Kulkarni-Nomizu product of the metric with itself; namely, if we denote

the (1,3)-curvature tensor and

the Riemann curvature tensor, where then

where is the scalar curvature and

is the Ricci tensor, which in components reads .

Proof. , thus

as is antisymmetric in the indices ; on the other hand,

since the Kulkarni-Nominzu product has the same symmetries of , as noticed before. Therefore, we conclude

For this very reason, it is commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor each makes to the curvature of a Riemannian manifold. This so-called Ricci decomposition is useful in differential geometry.

When there is a metric tensor g, the Kulkarni–Nomizu product of g with itself is the identity endomorphism of the space of 2-forms, Ω2(M), under the identification (using the metric) of the endomorphism ring End(Ω2(M)) with the tensor product Ω2(M)  Ω2(M).

A Riemannian manifold has constant sectional curvature k if and only if the Riemann tensor has the form

where g is the metric tensor.

Notes

  1. Some authors also include an overall factor in the definition.
  2. A (0,4)-tensor which satisfies the skew-symmetry property, the interchange symmetry property and the first (algebraic) Bianchi identity (see symmetries and identities of the Riemann curvature) is called an algebraic curvature tensor.

References

  • Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8.
  • Gallot, S., Hullin, D., and Lafontaine, J. (1990). Riemannian Geometry. Springer-Verlag.CS1 maint: multiple names: authors list (link)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.