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Changed the scope of the question since the original algebraic formulation had a simple answer and was not interesting.
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alvarezpaiva
alvarezpaiva
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A question in multilinear algebra: characterizing Characterizing maximal powers of closed 2-forms in odd-dimensional spacesmanifolds

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $\mathbb{R}^{2n+1}$$2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$?

Remark. This question started off as a question on multilinear algebra because I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is thethought that perhaps there was an algebraic point-wise condition on the numbers $A_i$ which are necessary and sufficient for$\Omega$ that was non-trivial, but that is not the existencecase: every element of a two form $\omega$ satisfying$\Lambda^{2n}(\mathbb{R}^{2n+1})$ is the $\Omega = \omega^n$$n$-th power of an element of $\Lambda^{2}(\mathbb{R}^{2n+1})$.

I feel this should be knownBackground. The odd dimensional manifolds in which I'm really interested are spherized tangent bundles (something neat such as Cartan's characterization of simplei.e. $k$-forms), so$STM := (TM \setminus 0)/\mathbb{R}^+$. The reason for this may be moreis that this question comes up in the study of a reference request. I would also be interestedinverse problems in knowing whether there is a way to know whether $\Omega$ is the productcalculus of $n$ (possibly different) $2$-formsvariations.

A question in multilinear algebra: characterizing maximal powers of 2-forms in odd-dimensional spaces

Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.

Characterizing maximal powers of closed 2-forms in odd-dimensional manifolds

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$?

Remark. This question started off as a question on multilinear algebra because I thought that perhaps there was an algebraic point-wise condition on $\Omega$ that was non-trivial, but that is not the case: every element of $\Lambda^{2n}(\mathbb{R}^{2n+1})$ is the $n$-th power of an element of $\Lambda^{2}(\mathbb{R}^{2n+1})$.

Background. The odd dimensional manifolds in which I'm really interested are spherized tangent bundles (i.e. $STM := (TM \setminus 0)/\mathbb{R}^+$. The reason for this is that this question comes up in the study of inverse problems in the calculus of variations.

added 742 characters in body; deleted 738 characters in body
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alvarezpaiva
alvarezpaiva
  • 13.5k
  • 40
  • 83

Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.

Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.

Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.

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alvarezpaiva
alvarezpaiva
  • 13.5k
  • 40
  • 83

A question in multilinear algebra: characterizing maximal powers of 2-forms in odd-dimensional spaces

Given a $2n$-form $\Omega$ in $\mathbb{R}^{2n+1}$, how do we know if there exists a $2$-form $\omega$ such that $\Omega = \omega^n$?

I would really like something very explicit. For example, if $$ \Omega = \sum_i A_i \ dx_1 \wedge \cdots \wedge \hat{dx_i} \wedge \cdots \wedge dx_{2n + 1} \ , $$ then what is the condition on the numbers $A_i$ which are necessary and sufficient for the existence of a two form $\omega$ satisfying $\Omega = \omega^n$.

I feel this should be known (something neat such as Cartan's characterization of simple $k$-forms), so this may be more of a reference request. I would also be interested in knowing whether there is a way to know whether $\Omega$ is the product of $n$ (possibly different) $2$-forms.