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| 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.
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| If ''h'' and ''k'' are symmetric (0,2)-tensors, then the product is defined via:
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| :<math>(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) </math>
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| where the ''X''<sub>''j''</sub> are tangent vectors.
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| Note that <math>h {~\wedge\!\!\!\!\!\!\bigcirc~} k = k {~\wedge\!\!\!\!\!\!\bigcirc~} h</math>. The Kulkarni–Nomizu product is a special case of the product in the graded algebra
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| :<math>\bigoplus_{p=1}^n S^2(\Omega^p M),</math>
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| where, on simple elements,
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| :<math>(\alpha\cdot\beta) {~\wedge\!\!\!\!\!\!\bigcirc~} (\gamma\cdot\delta) = (\alpha\wedge\gamma)\cdot(\beta\wedge\delta)</math>
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| (the dot denotes the symmetric product).
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| 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 Riemannian manifolds|curvature]] of a [[Riemannian manifold]]. This so-called [[Ricci decomposition]] is useful in [[differential geometry]].
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| 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, Ω<sup>''2''</sup>(''M''), under the identification (using the metric) of the endomorphism ring End(Ω<sup>''2''</sup>(''M'')) with the tensor product Ω<sup>''2''</sup>(''M'') ⊗ Ω<sup>''2''</sup>(''M'').
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| A Riemannian manifold has constant [[sectional curvature]] ''k'' if and only if the Riemann tensor has the form
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| :<math>R = \frac{k}{2}g {~\wedge\!\!\!\!\!\!\bigcirc~} g</math> | |
| where ''g'' is the [[metric tensor]].
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| ==References==
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| *{{Citation | last1=Besse | first1=Arthur L. | title=Einstein manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10 | isbn=978-3-540-15279-8 | year=1987 | pages=xii+510}}.
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| * {{cite book|author=Gallot, S., Hullin, D., and Lafontaine, J.|title=Riemannian Geometry|publisher=Springer-Verlag|year=1990}}
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| {{DEFAULTSORT:Kulkarni-Nomizu product}}
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| [[Category:Differential geometry]]
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| [[Category:Tensors]]
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| {{differential-geometry-stub}}
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