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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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Diffeomorphisms that let the Haar measure invariant and null divergent

Let $G$ be a compact Lie group with Haar measure $\mu$. Let $X\in\mathfrak{X} (G)$ be such that, if $T(x)=\exp_x(X_x)$, $$T_*\mu=\mu,$$ then $\operatorname{div}(X)=0$? This is true when $G=S^1$, becau …
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