Let $G$ be a finite group of isometries of a Riemannian manifold $\overline{M}$ acting properly discontinuously. Let $\pi : \overline{M} \to \overline{M}/G$ denote the projection onto the quotient. If $M$ is a connected hypersurface of the quotient, is it true that the number of connected components of $\pi^{-1}(M)$ is a divisor of the order of $G$?
2026-03-30 01:12:35.1774833155
Number of connected componnents
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Hint: Apply the orbit-stabilizer theorem. ("Stabilizer" here means "lands in the initial connected component".)