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

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.

$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$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"?

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.

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

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

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"?

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.

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

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.

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

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

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 became the name "Kulkarni-Nomizu product"?

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.

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

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

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"?

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.