Kosmann lift

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 $$\hat{X}\,$$ defined on the bundle of linear frames.

Generalisations exist for any given reductive G-structure.

Introduction
In general, given a subbundle $$Q\subset E\,$$ of a fiber bundle $$\pi_{E}\colon E\to M\,$$ over $$M$$ and a vector field $$Z\,$$ on $$E$$, its restriction $$Z\vert_Q\,$$ to $$Q$$ is a vector field "along" $$Q$$ not on (i.e., tangent to) $$Q$$. If one denotes by $$i_{Q} \colon Q\hookrightarrow E$$ the canonical embedding, then $$Z\vert_Q\,$$ is a section of the pullback bundle $$i^{\ast}_{Q}(TE) \to Q\,$$, where
 * $$i^{\ast}_{Q}(TE) = \{(q,v) \in Q \times TE \mid i(q) = \tau_{E}(v)\}\subset Q\times TE,\,$$

and $$\tau_{E}\colon TE\to E\,$$ is the tangent bundle of the fiber bundle $$E$$. Let us assume that we are given a  Kosmann decomposition of the pullback bundle $$i^{\ast}_{Q}(TE) \to Q\,$$, such that
 * $$i^{\ast}_{Q}(TE) = TQ\oplus \mathcal M(Q),\,$$

i.e., at each $$q\in Q$$ one has $$T_qE=T_qQ\oplus \mathcal M_u\,,$$ where $$\mathcal M_{u}$$ is a vector subspace of $$T_qE\,$$ and we assume $$\mathcal M(Q)\to Q\,$$ to be a vector bundle over $$Q$$, called the transversal bundle of the  Kosmann decomposition. It follows that the restriction $$Z\vert_Q\,$$ 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 $$\mathcal M(Q)\to Q.\,$$

Definition
Let $$\mathrm F_{SO}(M)\to M$$ be the oriented orthonormal frame bundle of an oriented $$n$$-dimensional Riemannian manifold $$M$$ with given metric $$g\,$$. This is a principal $${\mathrm S\mathrm O}(n)\,$$-subbundle of $$\mathrm FM\,$$, the tangent frame bundle of linear frames over $$M$$ with structure group $${\mathrm G\mathrm L}(n,\mathbb R)\,$$. By definition, one may say that we are given with a classical reductive $${\mathrm S\mathrm O}(n)\,$$-structure. The special orthogonal group $${\mathrm S\mathrm O}(n)\,$$ is a reductive Lie subgroup of $${\mathrm G\mathrm L}(n,\mathbb R)\,$$. In fact, there exists a direct sum decomposition $$\mathfrak{gl}(n)=\mathfrak{so}(n)\oplus \mathfrak{m}\,$$, where $$\mathfrak{gl}(n)\,$$ is the Lie algebra of $${\mathrm G\mathrm L}(n,\mathbb R)\,$$, $$\mathfrak{so}(n)\,$$ is the Lie algebra of $${\mathrm S\mathrm O}(n)\,$$, and $$\mathfrak{m}\,$$ is the $$\mathrm{Ad}_{\mathrm S\mathrm O}\,$$-invariant vector subspace of symmetric matrices, i.e. $$\mathrm{Ad}_{a}\mathfrak{m}\subset\mathfrak{m}\,$$ for all $$a\in{\mathrm S\mathrm O}(n)\,.$$

Let $$i_{\mathrm F_{SO}(M)} \colon \mathrm F_{SO}(M)\hookrightarrow \mathrm FM$$ be the canonical embedding. One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle $$i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM) \to \mathrm F_{SO}(M)$$ such that


 * $$i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM)=T\mathrm F_{SO}(M)\oplus \mathcal M(\mathrm F_{SO}(M))\,,$$

i.e., at each $$u\in \mathrm F_{SO}(M)$$ one has $$T_u\mathrm FM=T_u \mathrm F_{SO}(M)\oplus \mathcal M_u\,,$$ $$\mathcal M_{u}$$ being the fiber over $$u$$ of the subbundle $$\mathcal M(\mathrm F_{SO}(M))\to \mathrm F_{SO}(M)$$ of $$i^{\ast}_{\mathrm F_{SO}(M)}(V\mathrm FM) \to \mathrm F_{SO}(M)$$. Here, $$V\mathrm FM\,$$ is the vertical subbundle of $$T\mathrm FM\,$$ and at each $$u\in \mathrm F_{SO}(M)$$ the fiber $$\mathcal M_{u}$$ is isomorphic to the vector space of symmetric matrices $$\mathfrak{m}$$.

From the above canonical and equivariant decomposition, it follows that the restriction $$Z\vert_{\mathrm F_{SO}(M)}$$ of an $${\mathrm G\mathrm L}(n,\mathbb R)$$-invariant vector field $$Z\,$$ on $$\mathrm FM$$ to $$\mathrm F_{SO}(M)$$ splits into a $${\mathrm S\mathrm O}(n)$$-invariant vector field $$Z_{K}\,$$ on $$\mathrm F_{SO}(M)$$, 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 $$\hat{X}\vert_{\mathrm F_{SO}(M)}\,$$ to $$\mathrm F_{SO}(M)\to M$$ of its natural lift $$\hat{X}\,$$ onto $$\mathrm FM\to M$$ splits into a $${\mathrm S\mathrm O}(n)$$-invariant vector field $$X_{K}\,$$ on $$\mathrm F_{SO}(M)$$, called the Kosmann lift of $$X\,$$, and a transverse vector field $$X_{G}\,$$.