Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Closedness of the tangent spaces to the orbits of proper actions   In this note we show that for any proper action of a Banach--Lie group $G$ on
a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have
closed range for each $x \in M$, i.e., the tangent spaces of the orbits are
closed. As a consequence, for each free proper action on a Hilbert manifold,
the quotient $M/G$ carries a natural manifold structure.

18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
26
The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph