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 . 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.