Kulkarni–Nomizu product

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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

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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

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The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor.[2] For instance, on space forms (i.e. spaces of constant sectional curvature) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric   with itself; namely, if we denote by

 

the (1, 3)-curvature tensor and by

 

the Riemann curvature tensor with  , then

 

where   is the scalar curvature and

 

is the Ricci tensor, which in components reads  . Expanding the Kulkarni–Nomizu product   using the definition from above, one obtains

 

This is the same expression as stated in the article on the Riemann curvature tensor.

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

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  1. ^ Some authors include an overall factor 1/2 in the definition.
  2. ^ A (0, 4)-tensor that 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

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  • 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.{{cite book}}: CS1 maint: multiple names: authors list (link)