View source for One-parameter subgroup

''of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682001.png" /> over a normed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682002.png" />''

An analytic homomorphism of the additive group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682003.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682004.png" />, that is, an analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682005.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682006.png" /></td> </tr></table>

The image of this homomorphism, which is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682007.png" />, is also called a one-parameter subgroup. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682008.png" />, then the continuity of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682009.png" /> implies that it is analytic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820011.png" />, then for any tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820013.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820014.png" /> there exists a unique one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820015.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820016.png" /> as its tangent vector at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820017.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820020.png" /> is the [[Exponential mapping|exponential mapping]]. In particular, any one-parameter subgroup of the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820021.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820022.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820023.png" /> is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820024.png" /> are the geodesics passing through the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820025.png" />.

====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>