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.

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

$(h {~\wedge\!\!\!\!\!\!\bigcirc~} k)(X_1,X_2,X_3,X_4) := h(X_1,X_3)k(X_2,X_4) + h(X_2,X_4)k(X_1,X_3) - h(X_1,X_4)k(X_2,X_3) - h(X_2,X_3)k(X_1,X_4)$

where the X_j are tangent vectors.

Note that $h {~\wedge\!\!\!\!\!\!\bigcirc~} k = k {~\wedge\!\!\!\!\!\!\bigcirc~} h$. The Kulkarni–Nomizu product is a special case of the product in the graded algebra

$\bigoplus_{p=1}^n S^2(\Omega^p M),$

where, on simple elements,

$(\alpha\cdot\beta) {~\wedge\!\!\!\!\!\!\bigcirc~} (\gamma\cdot\delta) = (\alpha\wedge\gamma)\cdot(\beta\wedge\delta)$

(the dot denotes the symmetric product).

The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. It is thus 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

$R = \frac{k}{2}g {~\wedge\!\!\!\!\!\!\bigcirc~} g$

where g is the metric 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.