In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field X on a Riemannian manifold ( M , g ) is the canonical projection X K on the orthonormal frame bundle of its natural lift X ^ defined on the bundle of linear frames.
Generalisations exist for any given reductive G-structure.
In general, given a subbundle Q ⊂ E of a fiber bundle π E : E → M over M and a vector field Z on E , its restriction Z | Q to Q is a vector field "along" Q not on (i.e., tangent to) Q . If one denotes by i Q : Q ↪ E the canonical embedding, then Z | Q is a section of the pullback bundle i Q ∗ ( T E ) → Q , where
i Q ∗ ( T E ) = { ( q , v ) ∈ Q × T E ∣ i ( q ) = τ E ( v ) } ⊂ Q × T E , and τ E : T E → E is the tangent bundle of the fiber bundle E . Let us assume that we are given a Kosmann decomposition of the pullback bundle i Q ∗ ( T E ) → Q , such that
i Q ∗ ( T E ) = T Q ⊕ M ( Q ) , i.e., at each q ∈ Q one has T q E = T q Q ⊕ M u , where M u is a vector subspace of T q E and we assume M ( Q ) → Q to be a vector bundle over Q , called the transversal bundle of the Kosmann decomposition. It follows that the restriction Z | Q to Q splits into a tangent vector field Z K on Q and a transverse vector field Z G , being a section of the vector bundle M ( Q ) → Q .
Let F S O ( M ) → M be the oriented orthonormal frame bundle of an oriented n -dimensional Riemannian manifold M with given metric g . This is a principal S O ( n ) -subbundle of F M , the tangent frame bundle of linear frames over M with structure group G L ( n , R ) . By definition, one may say that we are given with a classical reductive S O ( n ) -structure. The special orthogonal group S O ( n ) is a reductive Lie subgroup of G L ( n , R ) . In fact, there exists a direct sum decomposition g l ( n ) = s o ( n ) ⊕ m , where g l ( n ) is the Lie algebra of G L ( n , R ) , s o ( n ) is the Lie algebra of S O ( n ) , and m is the A d S O -invariant vector subspace of symmetric matrices, i.e. A d a m ⊂ m for all a ∈ S O ( n ) .
Let i F S O ( M ) : F S O ( M ) ↪ F M be the canonical embedding.
One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle i F S O ( M ) ∗ ( T F M ) → F S O ( M ) such that
i F S O ( M ) ∗ ( T F M ) = T F S O ( M ) ⊕ M ( F S O ( M ) ) , i.e., at each u ∈ F S O ( M ) one has T u F M = T u F S O ( M ) ⊕ M u , M u being the fiber over u of the subbundle M ( F S O ( M ) ) → F S O ( M ) of i F S O ( M ) ∗ ( V F M ) → F S O ( M ) . Here, V F M is the vertical subbundle of T F M and at each u ∈ F S O ( M ) the fiber M u is isomorphic to the vector space of symmetric matrices m .
From the above canonical and equivariant decomposition, it follows that the restriction Z | F S O ( M ) of an G L ( n , R ) -invariant vector field Z on F M to F S O ( M ) splits into a S O ( n ) -invariant vector field Z K on F S O ( M ) , called the Kosmann vector field associated with Z , and a transverse vector field Z G .
In particular, for a generic vector field X on the base manifold ( M , g ) , it follows that the restriction X ^ | F S O ( M ) to F S O ( M ) → M of its natural lift X ^ onto F M → M splits into a S O ( n ) -invariant vector field X K on F S O ( M ) , called the Kosmann lift of X , and a transverse vector field X G .
