In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two -tensors and gives as a result a -tensor.
If h and k are symmetric -tensors, then the product is defined via:[1]
\begin{align} (h{~\wedge circ~}k)(X1,X2,X3,X4):={} &h(X1,X3)k(X2,X4)+h(X2,X4)k(X1,X3)\\ &{}-h(X1,X4)k(X2,X3)-h(X2,X3)k(X1,X4)\\[3pt] {}={}&\begin{vmatrix} h(X1,X3)&h(X1,X4)\\ k(X2,X3)&k(X2,X4) \end{vmatrix}+\begin{vmatrix} k(X1,X3)&k(X1,X4)\\ h(X2,X3)&h(X2,X4) \end{vmatrix} \end{align}
| ⋅ |
h{~\wedge circ~}k=k{~\wedge circ~}h
With respect to a basis
\{\partiali\}
(h~\wedge circ~k)ijlm=(h{~\wedge circ~}k)(\partiali,\partialj,\partiall,\partialm) =2hikm]j+2hjkl]i,
[...]
The Kulkarni–Nomizu product is a special case of the product in the graded algebra
n | |
oplus | |
p=1 |
S2\left(\OmegapM\right),
where, on simple elements,
(\alpha ⋅ \beta){~\wedge circ~}(\gamma ⋅ \delta)=(\alpha\wedge\gamma)\odot(\beta\wedge\delta)
(
\odot
g=gijdxi ⊗ dxj
\operatorname{R}(\partiali,\partialj)\partialk=
l} | |
{R | |
ijk |
\partiall
\operatorname{Rm}=Rijkldxi ⊗ dxj ⊗ dxk ⊗ dxl
Rijkl=gim
m} | |
{R | |
jkl |
\operatorname{Rm}= | \operatorname{Scal |
where
i} | |
\operatorname{Scal}=\operatorname{tr} | |
i |
\operatorname{Ric}(Y,Z)=\operatorname{tr}g\lbraceX\mapsto\operatorname{R}(X,Y)Z\rbrace
is the Ricci tensor, which in components reads
Rij
k} | |
={R | |
ikj |
g~\wedge circ~g
Rijkl=
\operatorname{Scal | |
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 (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
R=
k | |
2 |
g{~\wedge circ~}g