Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rigid geometric structures, isometric actions, and algebraic quotients   By using a Borel density theorem for algebraic quotients, we prove a theorem
concerning isometric actions of a Lie group $G$ on a smooth or analytic
manifold $M$ with a rigid $\mathrm{A}$-structure $\sigma$. It generalizes
Gromov's centralizer and representation theorems to the case where $R(G)$ is
split solvable and $G/R(G)$ has no compact factors, strengthens a special case
of Gromov's open dense orbit theorem, and implies that for smooth $M$ and
simple $G$, if Gromov's representation theorem does not hold, then the local
Killing fields on $\widetilde{M}$ are highly non-extendable. As applications of
the generalized centralizer and representation theorems, we prove (1) a
structural property of $\mathrm{Iso}(M)$ for simply connected compact analytic
$M$ with unimodular $\sigma$, (2) three results illustrating the phenomena that
if $G$ is split solvable and large then $\pi_1(M)$ is also large, and (3) two
fixed point theorems for split solvable $G$ and compact analytic $M$ with
non-unimodular $\sigma$.

21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int