Morphisms of geometric structures

The concept of $\{G,\rho,V\}$-structure is introduced which is a principal $G$-bundle B on which a $V$-valued form is given. If the representation $\rho$ of the group $G$ on the vector space $V$ is faithful and the fibration $B\to B\pmod G$ is locally trivial, then the $\{G,\rho,V\}$-structure is equivalent to some $G$-structure. The relation between local and global transitivity of the structure is studied under the condition that the space of the structure is compact and simply connected. It is proved that the universal covering space of a $\{G,\rho,V\}$-structure can be viewed as a $\{G',\rho',V\}$-structure.