One-parameter subgroup

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of a Lie group over a normed field

An analytic homomorphism of the additive group of the field into , that is, an analytic mapping such that

The image of this homomorphism, which is a subgroup of , is also called a one-parameter subgroup. If , then the continuity of the homomorphism implies that it is analytic. If or , then for any tangent vector to at the point there exists a unique one-parameter subgroup having as its tangent vector at the point . Here , , where is the exponential mapping. In particular, any one-parameter subgroup of the general linear group has the form

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


[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)



[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)
How to Cite This Entry:
One-parameter subgroup. r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098