  An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra
homomorphism $\zeta:\frak g\to \frak X(M)$. We define orbits for such an
action. In general the space of orbits $M/\frak g$ is not a manifold and even
has a bad topology. Nevertheless for a $\frak g$-manifold with equidimensional
orbits we treat such notions as connection, curvature, covariant
differentiation, Bianchi identity, parallel transport, basic differential
forms, basic cohomology, and characteristic classes, which generalize the
corresponding notions for principal $G$-bundles. As one of the applications, we
derive a sufficient condition for the projection $M\to M/\frak g$ to be a
bundle associated to a principal bundle.

Differential Geometry