Kosmann lift

In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field [math]\displaystyle{ X\, }[/math] on a Riemannian manifold [math]\displaystyle{ (M,g)\, }[/math] is the canonical projection [math]\displaystyle{ X_{K}\, }[/math] on the orthonormal frame bundle of its natural lift [math]\displaystyle{ \hat{X}\, }[/math] defined on the bundle of linear frames.

Generalisations exist for any given reductive G-structure.

Introduction

In general, given a subbundle [math]\displaystyle{ Q\subset E\, }[/math] of a fiber bundle [math]\displaystyle{ \pi_{E}\colon E\to M\, }[/math] over [math]\displaystyle{ M }[/math] and a vector field [math]\displaystyle{ Z\, }[/math] on [math]\displaystyle{ E }[/math], its restriction [math]\displaystyle{ Z\vert_Q\, }[/math] to [math]\displaystyle{ Q }[/math] is a vector field "along" [math]\displaystyle{ Q }[/math] not on (i.e., tangent to) [math]\displaystyle{ Q }[/math]. If one denotes by [math]\displaystyle{ i_{Q} \colon Q\hookrightarrow E }[/math] the canonical embedding, then [math]\displaystyle{ Z\vert_Q\, }[/math] is a section of the pullback bundle [math]\displaystyle{ i^{\ast}_{Q}(TE) \to Q\, }[/math], where

[math]\displaystyle{ i^{\ast}_{Q}(TE) = \{(q,v) \in Q \times TE \mid i(q) = \tau_{E}(v)\}\subset Q\times TE,\, }[/math]

and [math]\displaystyle{ \tau_{E}\colon TE\to E\, }[/math] is the tangent bundle of the fiber bundle [math]\displaystyle{ E }[/math]. Let us assume that we are given a Kosmann decomposition of the pullback bundle [math]\displaystyle{ i^{\ast}_{Q}(TE) \to Q\, }[/math], such that

[math]\displaystyle{ i^{\ast}_{Q}(TE) = TQ\oplus \mathcal M(Q),\, }[/math]

i.e., at each [math]\displaystyle{ q\in Q }[/math] one has [math]\displaystyle{ T_qE=T_qQ\oplus \mathcal M_u\,, }[/math] where [math]\displaystyle{ \mathcal M_{u} }[/math] is a vector subspace of [math]\displaystyle{ T_qE\, }[/math] and we assume [math]\displaystyle{ \mathcal M(Q)\to Q\, }[/math] to be a vector bundle over [math]\displaystyle{ Q }[/math], called the transversal bundle of the Kosmann decomposition. It follows that the restriction [math]\displaystyle{ Z\vert_Q\, }[/math] to [math]\displaystyle{ Q }[/math] splits into a tangent vector field [math]\displaystyle{ Z_K\, }[/math] on [math]\displaystyle{ Q }[/math] and a transverse vector field [math]\displaystyle{ Z_G,\, }[/math] being a section of the vector bundle [math]\displaystyle{ \mathcal M(Q)\to Q.\, }[/math]

Definition

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

Let [math]\displaystyle{ i_{\mathrm F_{SO}(M)} \colon \mathrm F_{SO}(M)\hookrightarrow \mathrm FM }[/math] be the canonical embedding.

One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle [math]\displaystyle{ i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM) \to \mathrm F_{SO}(M) }[/math] such that

[math]\displaystyle{ i^{\ast}_{\mathrm F_{SO}(M)}(T\mathrm FM)=T\mathrm F_{SO}(M)\oplus \mathcal M(\mathrm F_{SO}(M))\,, }[/math]

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

From the above canonical and equivariant decomposition, it follows that the restriction [math]\displaystyle{ Z\vert_{\mathrm F_{SO}(M)} }[/math] of an [math]\displaystyle{ {\mathrm G\mathrm L}(n,\mathbb R) }[/math]-invariant vector field [math]\displaystyle{ Z\, }[/math] on [math]\displaystyle{ \mathrm FM }[/math] to [math]\displaystyle{ \mathrm F_{SO}(M) }[/math] splits into a [math]\displaystyle{ {\mathrm S\mathrm O}(n) }[/math]-invariant vector field [math]\displaystyle{ Z_{K}\, }[/math] on [math]\displaystyle{ \mathrm F_{SO}(M) }[/math], called the Kosmann vector field associated with [math]\displaystyle{ Z\, }[/math], and a transverse vector field [math]\displaystyle{ Z_{G}\, }[/math].

In particular, for a generic vector field [math]\displaystyle{ X\, }[/math] on the base manifold [math]\displaystyle{ (M,g)\, }[/math], it follows that the restriction [math]\displaystyle{ \hat{X}\vert_{\mathrm F_{SO}(M)}\, }[/math] to [math]\displaystyle{ \mathrm F_{SO}(M)\to M }[/math] of its natural lift [math]\displaystyle{ \hat{X}\, }[/math] onto [math]\displaystyle{ \mathrm FM\to M }[/math] splits into a [math]\displaystyle{ {\mathrm S\mathrm O}(n) }[/math]-invariant vector field [math]\displaystyle{ X_{K}\, }[/math] on [math]\displaystyle{ \mathrm F_{SO}(M) }[/math], called the Kosmann lift of [math]\displaystyle{ X\, }[/math], and a transverse vector field [math]\displaystyle{ X_{G}\, }[/math].

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, 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3 
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993) (PDF), Natural operators in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf, 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