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Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms $A$, given a (smooth) matrix of two forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for smooth matrix of one-forms $B$ (i.e. the Bianchi identity is satisfied). Notice that this is true for line-bundles (in fact over any convex open set).

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $A$, given a (smooth) matrix of 2-forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for some smooth matrix of 1-forms $B$ (i.e. the Bianchi identity is satisfied). Notice that this is true for line-bundles (in fact over any convex open set).

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms $A$, given a (smooth) matrix of two forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for smooth matrix of one-forms $B$ (i.e. the Bianchi identity is satisfied). Notice that this is true for line-bundles (in fact over any convex open set). enter code here

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms $A$, given a (smooth) matrix of two forms $F$ which satisfies the condition $dF =B \wedge F - F \wedge B$ for smooth matrix of one-forms $B$ (i.e. the Bianchi identity is satisfied). Notice that this is true for line-bundles (in fact over any convex open set).