oneparameter subgroup
Let $G$ be a Lie Group^{}. A oneparameter subgroup of $G$ is a group homomorphism^{}
$$\varphi :\mathbb{R}\to G$$ 
that is also a differentiable map at the same time. We view $\mathbb{R}$ additively and $G$ multiplicatively, so that $\varphi (r+s)=\varphi (r)\varphi (s)$.
Examples.

1.
If $G=\mathrm{GL}(n,k)$, where $k=\mathbb{R}$ or $\u2102$, then any oneparameter subgroup has the form
$$\varphi (t)={e}^{tA},$$ where $A=\frac{d\varphi}{dt}(0)$ is an $n\times n$ matrix over $k$. The matrix $A$ is just a tangent vector to the Lie group $\mathrm{GL}(n,k)$. This property establishes the fact that there is a onetoone correspondence between oneparameter subgroups and tangent vectors of $\mathrm{GL}(n,k)$. The same relationship holds for a general Lie group. The onetoone correspondence between tangent vectors at the identity^{} (the Lie algebra) and oneparameter subgroups is established via the exponential map instead of the matrix exponential^{}.

2.
If $G=\mathrm{O}(n,\mathbb{R})\subseteq \mathrm{GL}(n,\mathbb{R})$, the orthogonal group^{} over $R$, then any oneparameter subgroup has the same form as in the example above, except that $A$ is skewsymmetric: ${A}^{\mathrm{T}}=A$.

3.
If $G=\mathrm{SL}(n,\mathbb{R})\subseteq \mathrm{GL}(n,\mathbb{R})$, the special linear group^{} over $R$, then any oneparameter subgroup has the same form as in the example above, except that $\mathrm{tr}(A)=0$, where $\mathrm{tr}$ is the trace operator.

4.
If $G=\mathrm{U}(n)=\mathrm{O}(n,\u2102)\subseteq \mathrm{GL}(n,\u2102)$, the unitary group^{} over $C$, then any oneparameter subgroup has the same form as in the example above, except that $A$ is skewHermitian (http://planetmath.org/SkewHermitianMatrix): $A={A}^{*}={\overline{A}}^{\mathrm{T}}$ and $\mathrm{tr}(A)=0$.
Title  oneparameter subgroup 

Canonical name  OneparameterSubgroup 
Date of creation  20130322 14:54:01 
Last modified on  20130322 14:54:01 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 22E15 
Classification  msc 22E10 
Synonym  1parameter subgroup 