Kulkarni–Nomizu product

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In the mathematical field of differential geometry, the Kulkarni–Nomizu product is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor. If h and k are (0,2)-tensors, then the product is defined via:

h\circ 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 Xj are tangent vectors.

It is most 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 conformal geometry.

[edit] References

  • Gallot, S., Hullin, D., and Lafontaine, J. (1990). Riemannian Geometry. Springer-Verlag. 
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