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I am trying to solve the equation $ $ \mathrm{d}G(x,y) = \mathrm{Vol}(x)+(-1)^{n+1}\mathrm{Vol}(y):= H $ $ for $ G\in \Omega^{n-1}\bigl((\mathbb{S}^n\times \mathbb{S}^n)\backslash \Delta=:M\bigr)$ . Here $ \mathrm{Vol}$ is the standard volume form on $ \mathbb{S}^n$ and $ \Delta$ the diagonal. The solution for $ n=1$ is the oriented angle $ \alpha(x,y)$ from $ x$ to $ y$Read more