1
$\begingroup$

Let $G$ be a compact Lie group with a biinvariant metric. Note that $G\times G$ acts isometrically on $G$ from left and right. Consider the quotient $D=G/H$ by a closed subgroup $H\le G\times G$; if $D$ can be obtained this way, then it will be called a double quotient of $G$.

Now suppose a group $L$ acts isometrically on the Hilbert space $\mathbb{H}$. Assume that the quotient $\mathbb{H}/L$ is isometric to a compact Riemannian manifold $M$.

Is it true that $M$ is isometric to a double quotient of a compact Lie group?

Comments

  • At the moment, I do not know if $M$ is analytic.

  • Any double quotient of a compact Lie group is a quotient of the Hilbert space by an isometric action (see “Submanifold geometry in symmetric spaces” by C.-L. Terng and G. Thorbergsson). In particular, the standard sphere can be obtained this way.

Improve this question
$\endgroup$
2
  • $\begingroup$ You make no assumptions about $L$ (abstract group, topological group, Lie group …), or about smoothness of the action? $\endgroup$
    – LSpice
    Commented Oct 20 at 20:51
  • 1
    $\begingroup$ @LSpice just a subgoup of group of isometries, but you can get something about this action since the quotient is compact Hausdorff space. $\endgroup$
    – Anton Petrunin
    Commented Oct 21 at 3:45

0

Reset to default

You must log in to answer this question.