Kosmann lift

In differential geometry, the Kosmann lift, ^{[1] [2]} named after Yvette Kosmann-Schwarzbach, of a vector field ${\displaystyle X\,}$ on a Riemannian manifold ${\displaystyle (M,g)\,}$ is the canonical projection ${\displaystyle X_{K}\,}$ on the orthonormal frame bundle of its natural lift ${\displaystyle {\hat {X}}\,}$ defined on the bundle of linear frames. ^[3]

Generalisations exist for any given reductive G-structure.

Introduction

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

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

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

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

i.e., at each ${\displaystyle q\in Q}$ one has ${\displaystyle T_{q}E=T_{q}Q\oplus {\mathcal {M}}_{u}\,,}$ where ${\displaystyle {\mathcal {M}}_{u}}$ is a vector subspace of ${\displaystyle T_{q}E\,}$ and we assume ${\displaystyle {\mathcal {M}}(Q)\to Q\,}$ to be a vector bundle over ${\displaystyle Q}$, called the transversal bundle of the Kosmann decomposition. It follows that the restriction ${\displaystyle Z\vert _{Q}\,}$ to ${\displaystyle Q}$ splits into a tangent vector field ${\displaystyle Z_{K}\,}$ on ${\displaystyle Q}$ and a transverse vector field ${\displaystyle Z_{G},\,}$ being a section of the vector bundle ${\displaystyle {\mathcal {M}}(Q)\to Q.\,}$

Definition

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

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

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

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

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

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

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

See also

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

Notes

1. Fatibene, L.; Ferraris, M.; Francaviglia, M.; Godina, M. (1996). "A geometric definition of Lie derivative for Spinor Fields". In Janyska, J.; Kolář, I.; Slovák, J. (eds.). Proceedings of the 6th International Conference on Differential Geometry and Applications, August 28th–September 1st 1995 (Brno, Czech Republic). Brno: Masaryk University. pp. 549–558. arXiv: gr-qc/9608003v1 . Bibcode:1996gr.qc.....8003F. ISBN   80-210-1369-9.
2. Godina, M.; Matteucci, P. (2003). "Reductive G-structures and Lie derivatives". Journal of Geometry and Physics . 47: 66–86. arXiv: math/0201235 . Bibcode:2003JGP....47...66G. doi:10.1016/S0393-0440(02)00174-2.
3. Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1, Wiley-Interscience, ISBN   0-470-49647-9 (Example 5.2) pp. 55-56

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