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Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\mathbb R}^n$ (for some $n$) sending $G$ to a subgroup of $O(n)$. I am interested in generalizations of this theorem in the case when the group $G$ is noncompact. In this setting, linear representations of $G$ should be replaced with projective representations, as it is done in the context of algebraic group actions on algebraic varieties:

Question:

  1. Suppose that $G$ is a connected noncompact linear Lie group acting smoothly on a smooth compact manifold $M$. Is there a smooth equivariant embedding $M\to {\mathbb R}P^{n-1}$ (for some $n$) sending $G$ to a subgroup of $PGL(n, {\mathbb R})$? (If it helps, let's assume that $G$ is semisimple; I would be also happy assuming that $M$ and $G$-action on $M$ are real-analytic.)

  2. Is there a topological analogue of such equivariant embedding theorem, where $M$ is a reasonable compact topological space (metrizable, of finite covering dimension) and $G$ is still a connected linear Lie group?

The only results in this direction I am aware of are in algebro-geometric setting and my guess is that there is a similar construction in the context of smooth manifolds, involving choice of a $G$-vector bundle $\xi: E\to M$ and a projective embedding of $M$ using smooth sections of $\xi$. I checked math.sci.net for papers referring to the ones by Mostow and Palais and got nothing interesting; google search also returned nothing of value.

Addendum: I realized that even an infinitesimal version of this question is unclear. Here is the infinitesimal question in holomorphic setting:

Question 3. Let ${\mathfrak g}$ be a finite-dimensional (complex) Lie subalgebra of the Lie algebra of holomorphic vector fields on $B^n$, the open complex $n$-ball. Is there (for some $m$) a holomorphic embedding $f: B^n \to {\mathbb C}P^{m-1}$ so that $f_*({\mathfrak g})$ is the restriction (to the image of $f$) of a subalgebra of the Lie algebra of linear vector fields $psl(m, {\mathbb C})$?

Somehow, this version seems much more plausible than the similar question about $C^\infty$ vector fields.

Note that, because of lack of compactness, the standard argument of constructing an embedding using sections of a suitable line bundle does not work. (Maybe one can impose some restrictions on holomorphic sections which would be preserved by the action of ${\mathfrak g}$ and would imply finite-dimensionality and separation of points.)

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    $\begingroup$ You seem to imply that any connected Lie group embeds (as a subgroup) into $PGL(n,\mathbb R)$. Does it really? $\endgroup$ – Igor Belegradek Nov 25 '13 at 12:37
  • $\begingroup$ @IgorBelegradek: Igor, you are right, I will add the linearity assumption for the Lie group. $\endgroup$ – Misha Nov 25 '13 at 13:45
  • $\begingroup$ As you probably aware, Mostow-Palais equivariant embedding also works for proper actions (of linear connected Lie groups). See [Kankaanrinta, "On embeddings of proper smooth G-manifolds", Math. Scand. 74 (1994)]. I would expect that without properness there are counterexamples. $\endgroup$ – Igor Belegradek Nov 25 '13 at 21:42
  • $\begingroup$ @IgorBelegradek: Igor, yes, I know the result about proper actions. However, I do not know how to prove or disprove the case of compact $M$ even in the case of action of the group of real numbers. $\endgroup$ – Misha Nov 26 '13 at 3:56

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