Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Question

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.)

Answer 1

There cannot be such an equivariant embedding in general. For a non-compact semisimple Lie group $G$ one can always find a cocompact lattice $\Gamma<G$, so $M=G/\Gamma$ will be a compact real analytic manifold on which $G$ acts real analytically. However, any homomorphism $G\to \text{PGL}_n(\mathbb{R})$ is algebraic, so the stabilizers of the corresponding action of $G$ on $\mathbb{P}^{n-1}(\mathbb{R})$ would be algebraic, thus you could not find a non-constant equivariant map $M\to \mathbb{P}^{n-1}(\mathbb{R})$, as $\Gamma$ is Zariski dense in $G$.

Answer 2

The following paper may be relevant:

ON EMBEDDINGS OF PROPER SMOOTH G-MANIFOLDS Author(s): MARJA KANKAANRINTASource: Mathematica Scandinavica, Vol. 74, No. 2 (1994), pp. 208-214 URL: https://www.jstor.org/stable/24490998