Let M = G/H be the homogeneous space of a Lie group G and a closed subgroup H. Denote by p : G → G/H the canonical projection, e ∈ G the identity and x0 = p(e). Let W be a subspace of the tangent space Tx0(M).
Definition. A lift W* of W is a subspace of the Lie algebra
of G satisfying
∩ W* = ﹛0﹜ and p*W* = W, where p* :
→ Tx0(M) denotes the tangent map of p at e.
Consider a G-invariant sub-bundle
of the tangent bundle of M (4), i.e., a field
of vector subspaces
x ⊂ Tx(M) for every x ∈ M satisfying
1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0004058X/resource/name/S0008414X0004058X_eqn01.gif?pub-status=live)
Here μg : M → M denotes the diffeomorphism defined by g ∈ G and (μg)*x : Tx → Tμg(x) the induced tangent map at x.