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:

where the X_j are tangent vectors.

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

where, on simple elements,

(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.

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