Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
2026-04-26 02:45:30.1777171530
Does map induced by rotation preserve the volume form?
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What is "the" volume form? If you work with the induced metric on $S^{n-1}$ from $\mathbb{R}^n$, then the map $A$ is an isometry (assuming $S^{n-1}$ is centered at the origin) and in particular, preserves the Riemannian volume form induced by the metric.