Kosmann lift

Template:Use dmy dates 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 ${\pi }_E:E\to M$ 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 $i_Q:Q\hookrightarrow E$ the canonical embedding, then $Z{|}_Q$ is a section of the pullback bundle $i_Q^{\ast }(TE)\to Q$, where

$i_Q^{\ast }(TE)=\{(q,v)\in Q\times TE\mid i(q)={\tau }_E(v)\}\subset Q\times TE,$

and ${\tau }_E:TE\to E$ is the tangent bundle of the fiber bundle $E$. Let us assume that we are given a Kosmann decomposition of the pullback bundle $i_Q^{\ast }(TE)\to Q$, such that

$i_Q^{\ast }(TE)=TQ\oplus ℳ(Q),$

i.e., at each $q\in Q$ one has $T_{q}E=T_{q}Q\oplus {ℳ}_u,$ where ${ℳ}_u$ is a vector subspace of $T_{q}E$ and we assume $ℳ(Q)\to Q$ to be a vector bundle over $Q$, called the transversal bundle of the Kosmann decomposition. It follows that the restriction $Z{|}_Q$ 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 $ℳ(Q)\to Q.$

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 $\mathrm{F}M$, the tangent frame bundle of linear frames over $M$ with structure group $\mathrm{G}\mathrm{L}(n,ℝ)$. 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 $\mathrm{G}\mathrm{L}(n,ℝ)$. In fact, there exists a direct sum decomposition $𝔤𝔩(n)=𝔰𝔬(n)\oplus 𝔪$, where $𝔤𝔩(n)$ is the Lie algebra of $\mathrm{G}\mathrm{L}(n,ℝ)$, $𝔰𝔬(n)$ is the Lie algebra of $\mathrm{S}\mathrm{O}(n)$, and $𝔪$ is the ${\mathrm{A}\mathrm{d}}_{\mathrm{S}\mathrm{O}}$-invariant vector subspace of symmetric matrices, i.e. ${\mathrm{A}\mathrm{d}}_a𝔪\subset 𝔪$ for all $a\in \mathrm{S}\mathrm{O}(n).$

Let $i_{{\mathrm{F}}_{SO}(M)}:{\mathrm{F}}_{SO}(M)\hookrightarrow \mathrm{F}M$ be the canonical embedding.

One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle $i_{{\mathrm{F}}_{SO}(M)}^{\ast }(T\mathrm{F}M)\to {\mathrm{F}}_{SO}(M)$ such that

$i_{{\mathrm{F}}_{SO}(M)}^{\ast }(T\mathrm{F}M)=T{\mathrm{F}}_{SO}(M)\oplus ℳ({\mathrm{F}}_{SO}(M)),$

i.e., at each $u\in {\mathrm{F}}_{SO}(M)$ one has $T_u\mathrm{F}M=T_u{\mathrm{F}}_{SO}(M)\oplus {ℳ}_u,$ ${ℳ}_u$ being the fiber over $u$ of the subbundle $ℳ({\mathrm{F}}_{SO}(M))\to {\mathrm{F}}_{SO}(M)$ of $i_{{\mathrm{F}}_{SO}(M)}^{\ast }(V\mathrm{F}M)\to {\mathrm{F}}_{SO}(M)$. Here, $V\mathrm{F}M$ is the vertical subbundle of $T\mathrm{F}M$ and at each $u\in {\mathrm{F}}_{SO}(M)$ 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 $Z{|}_{{\mathrm{F}}_{SO}(M)}$ of an $\mathrm{G}\mathrm{L}(n,ℝ)$-invariant vector field $Z$ on $\mathrm{F}M$ 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 $\hat{X}{|}_{{\mathrm{F}}_{SO}(M)}$ to ${\mathrm{F}}_{SO}(M)\to M$ of its natural lift $\hat{X}$ onto $\mathrm{F}M\to M$ 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 $X_G$.

See also

• Frame bundle
• Orthonormal frame bundle
• Principal bundle
• Spin bundle
• Connection (mathematics)
• G-structure
• Spin manifold
• Spin structure

References

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