In differential geometry, the Kosmann lift,[1][2] named after Yvette Kosmann-Schwarzbach, of a vector field
on the orthonormal frame bundle of its natural lift
defined on the bundle of linear frames.
[3] Generalisations exist for any given reductive G-structure.
and a vector field
is a vector field "along"
is a section of the pullback bundle
Let us assume that we are given a Kosmann decomposition of the pullback bundle
, called the transversal bundle of the Kosmann decomposition.
splits into a tangent vector field
and a transverse vector field
being a section of the vector bundle
be the oriented orthonormal frame bundle of an oriented
, the tangent frame bundle of linear frames over
By definition, one may say that we are given with a classical reductive
The special orthogonal group
is a reductive Lie subgroup of
In fact, there exists a direct sum decomposition
-invariant vector subspace of symmetric matrices, i.e.
One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle
is isomorphic to the vector space of symmetric matrices
From the above canonical and equivariant decomposition, it follows that the restriction
-invariant vector field
-invariant vector field
, called the Kosmann vector field associated with
, and a transverse vector field
In particular, for a generic vector field
-invariant vector field
, called the Kosmann lift of
, and a transverse vector field