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7 votes
1 answer
394 views

On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$. Is it true ...
asv's user avatar
  • 21.8k
3 votes
1 answer
337 views

Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely

Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...
Akerbeltz's user avatar
  • 516
4 votes
0 answers
132 views

Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
Eric Kubischta's user avatar
8 votes
1 answer
298 views

How special are homogeneous spaces?

Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$? Related questions/approaches: Of course we need $\...
GFR's user avatar
  • 639