View source for One-parameter subgroup

<!--
o0682001.png
$#A+1 = 25 n = 0
$#C+1 = 25 : ~/encyclopedia/old_files/data/O068/O.0608200 One\AAhparameter subgroup
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''of a Lie group  $  G $
over a normed field  $  K $''

An analytic homomorphism of the additive group of the field  $  K $
into  $  G $, 
that is, an analytic mapping  $  \alpha :  K \rightarrow G $
such that

$$ 
\alpha ( s + t)  =  \alpha ( s) \alpha ( t),\  s, t \in K.
$$

The image of this homomorphism, which is a subgroup of  $  G $, 
is also called a one-parameter subgroup. If  $  K = \mathbf R $, 
then the continuity of the homomorphism  $  \alpha :  K \rightarrow G $
implies that it is analytic. If  $  K = \mathbf R $
or  $  \mathbf C $, 
then for any tangent vector  $  X \in T _ {e} G $
to  $  G $
at the point  $  e $
there exists a unique one-parameter subgroup  $  \alpha :  K \rightarrow G $
having  $  X $
as its tangent vector at the point  $  t = 0 $. 
Here  $  \alpha ( t) =  \mathop{\rm exp}  tX $, 
$  t \in K $, 
where  $   \mathop{\rm exp} :  T _ {e} G \rightarrow G $
is the [[Exponential mapping|exponential mapping]]. In particular, any one-parameter subgroup of the [[General linear group|general linear group]]  $  G =  \mathop{\rm GL} ( n, K) $
has the form

$$ 
\alpha ( t)  =   \mathop{\rm exp}  tX  = \ 
\sum _ {n = 0 } ^  \infty  
{
\frac{1}{n! }
 }
t  ^ {n} X  ^ {n} .
$$

If  $  G $
is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of  $  G $
are the geodesics passing through the identity  $  e $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,   "Groupes et algèbres de Lie" , Hermann  (1972)  pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Hochschild,   "Structure of Lie groups" , Holden-Day  (1965)</TD></TR></table>