# One-parameter subgroup

*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 .

#### 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) |

#### Comments

#### 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) |

**How to Cite This Entry:**

One-parameter subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=One-parameter_subgroup&oldid=11235