<s>
In	O
differential	B-Language
geometry	I-Language
,	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
is	O
a	O
group	O
action	O
adapted	O
to	O
the	O
smooth	O
setting	O
:	O
G	O
is	O
a	O
Lie	O
group	O
,	O
M	O
is	O
a	O
smooth	O
manifold	O
,	O
and	O
the	O
action	O
map	O
is	O
differentiable	O
.	O
</s>
<s>
Let	O
be	O
a	O
(	O
left	O
)	O
group	O
action	O
of	O
a	O
Lie	O
group	O
G	O
on	O
a	O
smooth	O
manifold	O
M	O
;	O
it	O
is	O
called	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
(	O
or	O
smooth	O
action	O
)	O
if	O
the	O
map	O
is	O
differentiable	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
of	O
G	O
on	O
M	O
consists	O
of	O
a	O
Lie	O
group	O
homomorphism	O
.	O
</s>
<s>
A	O
smooth	O
manifold	O
endowed	O
with	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
is	O
also	O
called	O
a	O
G-manifold	O
.	O
</s>
<s>
Forgetting	O
the	O
smooth	O
structure	O
,	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
is	O
a	O
particular	O
case	O
of	O
a	O
continuous	O
group	O
action	O
.	O
</s>
<s>
For	O
every	O
Lie	O
group	O
G	O
,	O
the	O
following	O
are	O
Lie	B-Algorithm
group	I-Algorithm
actions	I-Algorithm
:	O
</s>
<s>
Other	O
examples	O
of	O
Lie	B-Algorithm
group	I-Algorithm
actions	I-Algorithm
include	O
:	O
</s>
<s>
Following	O
the	O
spirit	O
of	O
the	O
Lie	O
group-Lie	O
algebra	O
correspondence	O
,	O
Lie	B-Algorithm
group	I-Algorithm
actions	I-Algorithm
can	O
also	O
be	O
studied	O
from	O
the	O
infinitesimal	O
point	O
of	O
view	O
.	O
</s>
<s>
Indeed	O
,	O
any	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
induces	O
an	O
infinitesimal	O
Lie	O
algebra	O
action	O
on	O
M	O
,	O
i.e.	O
</s>
<s>
Conversely	O
,	O
by	O
Lie	O
–	O
Palais	O
theorem	O
,	O
any	O
abstract	O
infinitesimal	O
action	O
of	O
a	O
(	O
finite-dimensional	O
)	O
Lie	O
algebra	O
on	O
a	O
compact	O
manifold	O
can	O
be	O
integrated	O
to	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
an	O
infinitesimal	O
Lie	O
algebra	O
action	O
is	O
injective	O
if	O
and	O
only	O
if	O
the	O
corresponding	O
global	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
is	O
free	O
.	O
</s>
<s>
An	O
important	O
(	O
and	O
common	O
)	O
class	O
of	O
Lie	B-Algorithm
group	I-Algorithm
actions	I-Algorithm
is	O
that	O
of	O
proper	O
ones	O
.	O
</s>
<s>
Given	O
a	O
Lie	B-Algorithm
group	I-Algorithm
action	I-Algorithm
of	O
G	O
on	O
M	O
,	O
the	O
orbit	O
space	O
does	O
not	O
admit	O
in	O
general	O
a	O
manifold	O
structure	O
.	O
</s>
