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 $i_{Q}^{\ast }(TE)\to Q\,$, where

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

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

$i_{Q}^{\ast }(TE)=TQ\oplus {\mathcal {M}}(Q),\,$

i.e., at each $q\in Q$ one has $T_{q}E=T_{q}Q\oplus {\mathcal {M}}_{u}\,,$ where ${\mathcal {M}}_{u}$ is a vector subspace of $T_{q}E\,$ 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 $\mathrm {F} M\,$, 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 $i_{\mathrm {F} _{SO}(M)}\colon \mathrm {F} _{SO}(M)\hookrightarrow \mathrm {F} M$ 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

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

i.e., at each $u\in \mathrm {F} _{SO}(M)$ one has $T_{u}\mathrm {F} M=T_{u}\mathrm {F} _{SO}(M)\oplus {\mathcal {M}}_{u}\,,$ ${\mathcal {M}}_{u}$ being the fiber over $u$ of the subbundle ${\mathcal {M}}(\mathrm {F} _{SO}(M))\to \mathrm {F} _{SO}(M)$ of $i_{\mathrm {F} _{SO}(M)}^{\ast }(V\mathrm {F} M)\to \mathrm {F} _{SO}(M)$. 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 ${\mathcal {M}}_{u}$ 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 $\mathrm {F} M$ 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 ${\hat {X}}\vert _{\mathrm {F} _{SO}(M)}\,$ 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

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 30 March 2017, retrieved 4 June 2011
• Sternberg, S. (1983), Lectures on Differential Geometry (2nd ed.), New York: Chelsea Publishing Co., ISBN 0-8218-1385-4
• Fatibene, Lorenzo; Francaviglia, Mauro (2003), Natural and Gauge Natural Formalism for Classical Field Theories, Kluwer Academic Publishers, ISBN 978-1-4020-1703-2