n} :J\to \mathrm{SO(n-1)}$$ are smooth maps and $X(r,x,y,R)$ is as in (\[gauge\]). We put $$m=^t(m^2,\dots,m^n)=\frac{\rho}{r}^t(R^2_j,\dots, R^n_j).$$ It is now an easy matter to check that the variation $$\begin{split}
\gamma_{\tau}(u) &= \mathbf{j}\circ(\alpha|_u+ \tau\sum_{j=2}^n m^j(u)\mathbf{b}_j|_u), \quad
(u,\tau)\in J\times (-\epsilon,\epsilon),\\
\gamma_{\tau}(u) &=\gamma(u),\quad  (u,\tau)\in (I-J)\times (-\epsilon,\epsilon),
\end{split}$$ satisfies the required properties. This concludes the proof of the Theorem.

The next theorem shows that conformal world-lines lie in an Einstein universe of dimension 2,3 or 4.

\[ThmB\][The trajectory of a conformal world-line is contained in a $m$-dimensional Lorentzian cycle, with $2\le m \le 4$.]{}

Let $\gamma$ be a conformal world-line parameterized by conformal parameter. We consider a second-order frame field $A=(A_0,\dots A_{n+1})$ along $\gamma$. Then $$\mathcal{T}^5(\gamma)|_u=\mathrm{Span}(A_0|_u,A_1|_u,A_2|_u,S_{\gamma}|_u,DS_{\gamma}|_u,A_{n+1}|_u),$$ for every $u\in I$. From (\[LS\]) we have $$\label{STR1}\begin{split}
\dot{A}_0&=A_1,\quad \dot{A}_1=-h_1A_0-A_{n+1},\\
\dot{A}_2&=A_0+S_{\gamma},\quad \dot{A}_{n+1}=-h_1A_1+A_2,
\end{split}$$ and $$\label{STR2}
\dot{A}_j=-s_jA_2+\sum_{k=3}^{n}\phi_j^kA_k,\quad  j=3,\dots, n.$$ The first equation in (\[EL1\]) implies $$\label{EL1Bis}(h_2)^2=\frac{2}{3}h_1+c_1,\quad c_1\in \R.$$ The remaining equations in (\[EL1\]) can be written as $$\label{EL2}
\dot{s}=-\phi\cdot s+s_{(1)},\quad \dot{s}_{(1)}=-(\frac{4}{3}h_1-c_1)s-\phi\cdot s_{(1)}$$ or, equivalently, in the form $$\label{EL3}\begin{cases}
\frac{d}{du}(S_{\gamma})=DS_{\gamma}-(\frac{2}{3}h_1+c_1)A_2,\\ \frac{d}{du}(DS|_{\gamma})=-(\frac{4}{3}h_1-c_1)S_{\gamma}-\frac{1}{3}\dot{h}_1A_2.
\end{cases}$$ From (\[EL2\]) we see that $s\wedge s_{(1)}$ is a solution of the linear system $$\frac{d}{du}(s\wedge s_{(1)}) = -(\phi\cdot s)\wedge s_{(1)}- s\wedge (\phi\cdot s_{(1)}).$$ Hence, two possibilities may occur :

-   Case I : $s|_u\wedge s_{(1)}|_u\neq 0$, for every 