Geodesic
Invariant Sub-Bundles of the Tangent Bundle of a Homogeneous Space
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 629-634

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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 * : → T x0(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 Here μg : M → M denotes the diffeomorphism defined by g ∈ G and (μg)*x : Tx → Tμg(x) the induced tangent map at x. 
Tondeur, Philippe. Invariant Sub-Bundles of the Tangent Bundle of a Homogeneous Space. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 629-634. doi: 10.4153/CJM-1966-062-0
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