Kosmann lift
In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field  on a Riemannian manifold  is the canonical projection  on the orthonormal frame bundle of its natural lift  defined on the bundle of linear frames. Generalisations exist for any given reductive G-structure. Introduction
In general, given a subbundle  of a fiber bundle  over  and a vector field  on , its restriction  to  is a vector field "along"  not on (i.e., tangent to) . If one denotes by  the canonical embedding, then  is a section of the pullback bundle , where

and  is the tangent bundle of the fiber bundle . Let us assume that we are given a  Kosmann decomposition of the pullback bundle ,  such that

i.e., at each  one has  where  is a vector subspace of   and we assume  to be a vector bundle over , called the transversal bundle of the  Kosmann decomposition. It follows that the restriction  to  splits into a tangent vector field  on  and a transverse vector field  being a section of the vector bundle

Definition

Let  be the oriented orthonormal frame bundle of an oriented -dimensional 
Riemannian manifold  with given metric . This is a principal -subbundle of , the tangent frame bundle of linear frames over  with structure group . By definition, one may say that we are given with a classical reductive -structure. The special orthogonal group  is a reductive Lie subgroup of . In fact, there exists a direct sum decomposition , where  is the Lie algebra of ,  is the Lie algebra of , and  is the -invariant vector subspace of symmetric matrices, i.e. for all 

Let  be the canonical embedding. One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle  such that

i.e., at each  one has   being the fiber over  of the subbundle  of . Here,  is the vertical subbundle of  and at each  the fiber  is isomorphic to the vector space of symmetric matrices . From the above canonical and equivariant decomposition, it follows that the restriction  of an -invariant vector field  on  to  splits into a -invariant vector field  on , called the Kosmann vector field associated with , and a transverse vector field . In particular, for a generic vector field  on the base manifold , it follows that the restriction  to  of its natural lift  onto  splits into a -invariant vector field  on , called the Kosmann lift of , and a transverse vector field . See also
 Frame bundle
 Orthonormal frame bundle
 Principal bundle
 Spin bundle
 Connection (mathematics)
 G-structure
 Spin manifold
 Spin structure

Notes

References
 
 

 

Fiber bundles
Vector bundles
Riemannian geometry
Structures on manifolds