Kosmann lift

In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field formula_1 on a Riemannian manifold formula_2 is the canonical projection formula_3 on the orthonormal frame bundle of its natural lift formula_4 defined on the bundle of linear frames.
Generalisations exist for any given reductive G-structure.
Introduction.
In general, given a subbundle formula_5 of a fiber bundle formula_6 over formula_7 and a vector field formula_8 on formula_9, its restriction formula_10 to formula_11 is a vector field "along" formula_11 not "on" (i.e., "tangent" to) formula_11. If one denotes by formula_14 the canonical embedding, then formula_10 is a section of the pullback bundle formula_16, where
and formula_18 is the tangent bundle of the fiber bundle formula_9.
Let us assume that we are given a Kosmann decomposition of the pullback bundle formula_16, such that
i.e., at each formula_22 one has formula_23 where formula_24 is a vector subspace of formula_25 and we assume formula_26 to be a vector bundle over formula_11, called the transversal bundle of the Kosmann decomposition. It follows that the restriction formula_10 to formula_11 splits into a "tangent" vector field formula_30 on formula_11 and a "transverse" vector field formula_32 being a section of the vector bundle formula_33
Definition.
Let formula_34 be the oriented orthonormal frame bundle of an oriented formula_35-dimensional 
Riemannian manifold formula_7 with given metric formula_37. This is a principal formula_38-subbundle of formula_39, the tangent frame bundle of linear frames over formula_7 with structure group formula_41.
By definition, one may say that we are given with a classical reductive formula_38-structure. The special orthogonal group formula_38 is a reductive Lie subgroup of formula_41. In fact, there exists a direct sum decomposition formula_45, where formula_46 is the Lie algebra of formula_41, formula_48 is the Lie algebra of formula_38, and formula_50 is the formula_51-invariant vector subspace of symmetric matrices, i.e. formula_52 for all formula_53
Let formula_54 be the canonical embedding.
One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle formula_55 such that
i.e., at each formula_57 one has formula_58 formula_24 being the fiber over formula_60 of the subbundle formula_61 of formula_62. Here, formula_63 is the vertical subbundle of formula_64 and at each formula_57 the fiber formula_24 is isomorphic to the vector space of symmetric matrices formula_67.
From the above canonical and equivariant decomposition, it follows that the restriction formula_68 of an formula_69-invariant vector field formula_8 on formula_71 to formula_72 splits into a formula_73-invariant vector field formula_74 on formula_72, called the Kosmann vector field associated with formula_8, and a "transverse" vector field formula_77.
In particular, for a generic vector field formula_1 on the base manifold formula_2, it follows that the restriction formula_80 to formula_34 of its natural lift formula_4 onto formula_83 splits into a formula_73-invariant vector field formula_3 on formula_72, called the Kosmann lift of formula_1, and a "transverse" vector field formula_88.