In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field X {\displaystyle X\,} on a Riemannian manifold ( M , g ) {\displaystyle (M,g)\,} is the canonical projection X K {\displaystyle X_{K}\,} on the orthonormal frame bundle of its natural lift X ^ {\displaystyle {\hat {X}}\,} defined on the bundle of linear frames. Generalisations exist for any given reductive G-structure.
Introduction In general, given a subbundle Q ⊂ E {\displaystyle Q\subset E\,} of a fiber bundle π E : E → M {\displaystyle \pi _{E}\colon E\to M\,} over M {\displaystyle M} and a vector field Z {\displaystyle Z\,} on E {\displaystyle E} , its restriction Z | Q {\displaystyle Z\vert _{Q}\,} to Q {\displaystyle Q} is a vector field "along" Q {\displaystyle Q} not on (i.e., tangent to) Q {\displaystyle Q} . If one denotes by i Q : Q ↪ E {\displaystyle i_{Q}\colon Q\hookrightarrow E} the canonical embedding, then Z | Q {\displaystyle Z\vert _{Q}\,} is a section of the pullback bundle i Q ∗ ( T E ) → Q {\displaystyle i_{Q}^{\ast }(TE)\to Q\,} , where
i Q ∗ ( T E ) = { ( q , v ) ∈ Q × T E ∣ i ( q ) = τ E ( v ) } ⊂ Q × T E , {\displaystyle i_{Q}^{\ast }(TE)=\{(q,v)\in Q\times TE\mid i(q)=\tau _{E}(v)\}\subset Q\times TE,\,}
and τ E : T E → E {\displaystyle \tau _{E}\colon TE\to E\,} is the tangent bundle of the fiber bundle E {\displaystyle E} . Let us assume that we are given a Kosmann decomposition of the pullback bundle i Q ∗ ( T E ) → Q {\displaystyle i_{Q}^{\ast }(TE)\to Q\,} , such that
i Q ∗ ( T E ) = T Q ⊕ M ( Q ) , {\displaystyle i_{Q}^{\ast }(TE)=TQ\oplus {\mathcal {M}}(Q),\,}
i.e., at each q ∈ Q {\displaystyle q\in Q} one has T q E = T q Q ⊕ M u , {\displaystyle T_{q}E=T_{q}Q\oplus {\mathcal {M}}_{u}\,,} where M u {\displaystyle {\mathcal {M}}_{u}} is a vector subspace of T q E {\displaystyle T_{q}E\,} and we assume M ( Q ) → Q {\displaystyle {\mathcal {M}}(Q)\to Q\,} to be a vector bundle over Q {\displaystyle Q} , called the transversal bundle of the Kosmann decomposition. It follows that the restriction Z | Q {\displaystyle Z\vert _{Q}\,} to Q {\displaystyle Q} splits into a tangent vector field Z K {\displaystyle Z_{K}\,} on Q {\displaystyle Q} and a transverse vector field Z G , {\displaystyle Z_{G},\,} being a section of the vector bundle M ( Q ) → Q . {\displaystyle {\mathcal {M}}(Q)\to Q.\,}
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