# 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.

## Definition

If h and k are symmetric (0,2)-tensors, then the product is defined via[1]:

{\displaystyle {\begin{aligned}(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})\\&={\begin{vmatrix}h(X_{1},X_{3})&h(X_{1},X_{4})\\k(X_{2},X_{3})&k(X_{2},X_{4})\end{vmatrix}}+{\begin{vmatrix}k(X_{1},X_{3})&k(X_{1},X_{4})\\h(X_{2},X_{3})&h(X_{2},X_{4})\end{vmatrix}}\end{aligned}}}

where the Xj are tangent vectors and ${\displaystyle |\cdot |}$ is the matrix determinant. Note that ${\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k=k{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}h}$ , as it is clear from the second expression.

With respect to a basis ${\displaystyle \{\partial _{i}\}}$ of the tangent space, it takes the compact form

${\displaystyle (h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}k)(\partial _{i},\partial _{j},\partial _{l},\partial _{m})=2h_{i[m}k_{l]j}+2h_{j[l}k_{m]i}=(h~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~k)_{ijlm},}$

where ${\displaystyle [\dots ]}$ denotes the total antisymmetrisation symbol.

The Kulkarni–Nomizu product is a special case of the product in the graded algebra

${\displaystyle \bigoplus _{p=1}^{n}S^{2}(\Omega ^{p}M),}$

where, on simple elements,

${\displaystyle (\alpha \cdot \beta ){~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}(\gamma \cdot \delta )=(\alpha \wedge \gamma )\odot (\beta \wedge \delta )}$

(${\displaystyle \odot }$ denotes the symmetric product).

## Properties

The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor[2]. For instance, on a 2-dimensional smooth Riemannian manifold, the Riemann curvature tensor has a simple expression in terms of the Kulkarni-Nomizu product of the metric ${\displaystyle g=g_{ij}dx^{i}\otimes dx^{j}}$ with itself; namely, if we denote

${\displaystyle \operatorname {R} (\partial _{i},\partial _{j})\partial _{k}={R_{ijk}}^{l}\partial _{l}}$

the (1,3)-curvature tensor and

${\displaystyle \operatorname {Rm} =R_{ijkl}dx^{i}\otimes dx^{j}\otimes dx^{k}\otimes dx^{l}}$

the Riemann curvature tensor, where ${\displaystyle R_{ijkl}={R_{ijk}}^{m}g_{ml}}$ then

${\displaystyle \operatorname {Rm} ={\frac {\operatorname {Scal} }{4}}g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g,}$

where ${\displaystyle \operatorname {Scal} =\operatorname {tr} _{g}\operatorname {Ric} ={R^{i}}_{i}}$ is the scalar curvature and

${\displaystyle \operatorname {Ric} (Y,Z)=\operatorname {tr} _{g}\lbrace X\mapsto \operatorname {R} (X,Y)Z\rbrace }$

is the Ricci tensor, which in components reads ${\displaystyle R_{ij}={R_{lij}}^{l}}$ .

Proof. ${\displaystyle \operatorname {Scal} ={R^{i}}_{i}=g^{ij}R_{ij}=g^{ij}g^{lm}R_{lijm}}$ , thus

${\displaystyle R_{lijm}=g_{ij}g_{lm}\operatorname {Scal} =\operatorname {Scal} (g_{m[l}g_{i]j}+g_{m(l}g_{i)j})=\operatorname {Scal} (g_{m[l}g_{i]j}),}$

as ${\displaystyle R_{lijm}}$ is antisymmetric in the indices ${\displaystyle l,i}$ ; on the other hand,

${\displaystyle (g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g)_{mjil}=4g_{m[l}g_{i]j}=(g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g)_{ilmj}=(g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g)_{lijm},}$

since the Kulkarni-Nominzu product has the same symmetries of ${\displaystyle \operatorname {Rm} }$ , as noticed before. Therefore, we conclude

${\displaystyle (\operatorname {Rm} )_{lijm}={\frac {\operatorname {Scal} }{4}}(g~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~g)_{lijm}.}$

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

${\displaystyle R={\frac {k}{2}}g{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc ~}g}$

where g is the metric tensor.

## Notes

1. Some authors also include an overall factor ${\displaystyle {\frac {1}{2}}}$ in the definition.
2. A (0,4)-tensor ${\displaystyle T}$ which satisfies the skew-symmetry property, the interchange symmetry property and the first (algebraic) Bianchi identity (see symmetries and identities of the Riemann curvature) is called an algebraic curvature tensor.

## References

• Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8.
• Gallot, S., Hullin, D., and Lafontaine, J. (1990). Riemannian Geometry. Springer-Verlag.CS1 maint: multiple names: authors list (link)