One-parameter subgroup

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. In particular, any one-parameter subgroup of the 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

[1]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[2]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[3]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)

References

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