Etymology “Kulkarni–Nomizu product”

Question

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two symmetric $2$-tensor fields $T,S\in\Gamma^{\infty}(T^{\ast}\mathcal{M}^{\otimes_{s}2})$ and produces a covariant $4$-tensor field $T \KN S$ by $$(T\KN S)(X_{1},X_{2},X_{3},X_{4}):=T(X_{1},X_{3})S(X_{2},X_{4})+T(X_{2},X_{4})S(X_{1},X_{3})-T(X_{1},X_{4})S(X_{2},X_{3})-T(X_{2},X_{3})S(X_{1},X_{4})$$ for all $X_{i}\in\mathfrak{X}(\mathcal{M})$, which has the same symmetry properties as the Riemannian curvature tensor.

Now, out of curiosity, I wanted to look up in which context the product was first discussed. If I am not mistaken, it seems that the name goes back to the work of R. S. Kulkarni and K. Nomizu, which introduced similar products in their work on double forms in the 1970s (see below), however, I am not able to understand the precise relation. A "double form" is an element of the tensor product

$$\mathcal{D}^{p,q}(\mathcal{M}):=\Omega^{p}(\mathcal{M})\otimes_{C^{\infty}(\mathcal{M})}\Omega^{q}(\mathcal{M}),$$

i.e. a section of the bundle $\bigwedge^{p}T^{\ast}\mathcal{M}\otimes\bigwedge^{q}T^{\ast}\mathcal{M}$. One then defines the direct sum

$$\mathcal{D}(\mathcal{M})=\bigoplus_{p,q=0}^{\infty}\mathcal{D}^{p,q}(\mathcal{M}).$$

This space can be given the structure of an anticommutative, associative bi-graded algebra, whose multiplication $\cdot$, called the "exterior product", is for pure tensors $\omega_{1}=\theta_{1}\otimes\theta_{2}\in\mathcal{D}^{p,q}(\mathcal{M})$ and $\omega_{2}=\theta_{3}\otimes\theta_{4}\in\mathcal{D}^{r,s}(\mathcal{M})$ given by

$$\omega_{1}\cdot\omega_{2}:=(\theta_{1}\wedge\theta_{3})\otimes (\theta_{2}\wedge\theta_{4})\in\mathcal{D}^{p+r,q+s}(\mathcal{M}).$$

It is anticommutative in the sense that $\omega_{1}\cdot\omega_{2}=(-1)^{pr+qs}\omega_{2}\cdot\omega_{1}$. Now, can anyone help me to fill in the gap and explain how this product is related to the Kulkarni–Nomizu product? Does anyone have more historical insight of how the product for symmetric tensor fields defined above in the context of Riemannian geometry came to the name "Kulkarni–Nomizu product"?

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The two original papers by Kulkarni and Nomizu are

• R. S. Kulkarni. On the Bianchi identities. Mathematische Annalen, vol. 199, num. 4, pages 175–204, 1972.

• K. Nomizu. On the decomposition of generalized curvature tensor fields. Codazzi, Ricci, Bianchi and Weyl revisited. In Differential Geometry. In Honor of Kentaro Yano, pages 335–345. Kinokuniya, Tokyo, 1972.

Answer 1

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$If $\omega_{1},\omega_{2}\in \mathcal{D}^{1,1}(M)$ are symmetric, then $\omega_{1}\cdot\omega_{2}$ coincides with the Kulkarni–Numizu product of $\omega_{1}$ and $\omega_{2}$ (maybe up to a sign, depending on the convention). This can be shown by observing directly that both $\omega_{1}\cdot\omega_{2}$ and $\omega_{1}\KN\omega_{2}$ are defined as the double alternation of $\omega_{1}\otimes\omega_{2}$: one first apply alternation on the first two indices, and then on the last two indices.

The "wedge" product for double forms is more general than the Kulkarni–Nomizu product, and makes sense even if the double forms are not symmetric. What Kulkarni showed in his paper is that there is a bundle map of double forms known as the "Bianchi sum" (which he denotes by $\mathfrak{G}$) which satisfies a Leibniz rule with respect to the wedge product. Then, the elements of $\ker\mathfrak{G}$ can be interpreted to satisfy a generalized algebraic Bianchi identity. For $\ker\mathfrak{G}\cap\mathcal{D}^{1,1}(M)$, these are exactly the symmetric tensors, and for $\ker\mathfrak{G}\cap\mathcal{D}^{2,2}(M)$, these are exactly the tensors satisfying the algebraic Bianchi identity of the curvature tensor.

By the way, double forms and this graded structure, along with the generalized Bianchi identity, were introduced even before Kulkarni's paper. See for example Calabi's paper from 1961 (On compact, Riemannian manifolds with constant curvature) or even de Rham's paper (Differentiable manifolds).

I don't know the history of the term for certain, but I guess that the term "Kulkarni–Nomizu product" came up specifically for the product on symmetric tensors since it is the most useful one (e.g. for the decomposition of curvature tensors, which by the way Kulkarni proves for $\ker\mathfrak{G}\cap\mathcal{D}^{k,k}(M)$ in general), and does not require the introduction of double forms in order to be of use.