Kulkarni–Nomizu product
In the mathematical field of differential geometry, the Kulkarni–Nomizu product is defined for two -tensors and gives as a result a -tensor.Definition
If h and k are symmetric -tensors, then the product is defined via:
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,
.Properties
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. For instance, on space forms 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 -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 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, under the identification of the endomorphism ring End with the tensor product Ω2 ⊗ Ω2.
A Riemannian manifold has constant sectional curvature k if and only if the Riemann tensor has the form
where g is the metric tensor.
