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: where the Xj are tangent vectors and |\cdot| 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 [\dots] denotes the total antisymmetrisation symbol. The Kulkarni–Nomizu product is a special case of the product in the graded algebra where, on simple elements, (\odot denotes the symmetric product).

Properties

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