Let $H$ be a Lie group and $G$ a Lie subgroup of $H$ . Denote by $\mathfrak{h}$ the Lie algebra of $H$ and by $\mathfrak{g}$ the Lie algebra of $G$ . We shall say that $G$ is a reductive Lie subgroup of $H$ if there exists a direct sum decomposition

$$\mathfrak{h}=\mathfrak{g}\oplus\mathfrak{m},$$

where $\mathfrak{m}$ is an $\mathrm{Ad}_{G}$ -invariant vector subspace of $\mathfrak{h}$ , i.e. $\mathrm{Ad}_{a}(\mathfrak{m})\subset\mathfrak{m}$ for all $a\in G$ (which means that the $\mathrm{Ad}_{G}$ representation of $G$ in $\mathfrak{h}$ is reducible into a direct sum decomposition of two $\mathrm{Ad}_{G}$ -invariant vector spaces: cf. [ 18 ] , p. $83$ ).

Let $P(M,G)$ be a principal bundle and $\rho\colon\Gamma\to G$ a central homomorphism of a Lie group $\Gamma$ onto $G$ , i.e. such that its kernel is discrete and contained in the centre of $\Gamma$ [ 14 ] (see also [ 15 ] ). A $\Gamma$ -structure on $P(M,G)$ is a principal bundle map $\zeta\colon\tilde{P}\to P$ which is equivariant under the right actions of the structure groups, i.e.

$$\zeta(\tilde{u}\cdot\alpha)=\zeta(\tilde{u})\cdot\rho(\alpha)$$

for all $\tilde{u}\in\tilde{P}$ and $\alpha\in\Gamma$ .

If $k,m$ are positive integers, we define their unitary product as

$$k\oplus m=\begin{cases}km&\gcd(k,m)=1\\ 0&\text{ otherwise}\end{cases}$$

(4)

If $k\oplus m=p$ , then we write ${k\left\lvert\lvert{p}\right.}$ and say that $k$ is a unitary divisor of $p$ .

A smooth curve $c:I\rightarrow N$ , is called a $\rho$ -admissible curve if

$$(\rho\circ c)(t)=\dot{\tilde{c}}(t)\,,$$

for all $t\in I$ . The projection $\tilde{c}$ of a $\rho$ -admissible curve will be called a base curve.