Kosmann lift

In differential geometry, the Kosmann lift,^[1][2][3] named after Yvette Kosmann-Schwarzbach, of a vector field $X\,$ 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.^[4]

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 $M$ and a vector field $Z\,$ on $E$, its restriction ${\displaystyle Z\vert _{Q}\,}$ to $Q$ is a vector field "along" $Q$ not on (i.e., tangent to) $Q$. 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 $E$. 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 $q\in Q$ 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 $Q$, called the transversal bundle of the Kosmann decomposition. It follows that the restriction ${\displaystyle Z\vert _{Q}\,}$ to $Q$ splits into a tangent vector field ${\displaystyle Z_{K}\,}$ on $Q$ and a transverse vector field ${\displaystyle Z_{G},\,}$ being a section of the vector bundle ${\displaystyle {\mathcal {M}}(Q)\to Q.\,}$

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 ${\displaystyle {\mathrm {S} \mathrm {O} }(n)\,}$-subbundle of ${\mathrm F}M\,$, the tangent frame bundle of linear frames over $M$ 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 $u$ 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 ${\mathfrak {m}}$.

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 $Z\,$ 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 $Z\,$, and a transverse vector field ${\displaystyle Z_{G}\,}$.

In particular, for a generic vector field $X\,$ on the base manifold ${\displaystyle (M,g)\,}$, it follows that the restriction ${\displaystyle {\hat {X}}\vert _{\mathrm {F} _{SO}(M)}\,}$ to $\mathrm F_{SO}(M)\to M$ 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 $X\,$, 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. and Godina M. (1996), A geometric definition of Lie derivative for Spinor Fields, in: Proceedings of the 6th International Conference on Differential Geometry and Applications, August 28th–September 1st 1995 (Brno, Czech Republic), Janyska J., Kolář I. & J. Slovák J. (Eds.), Masaryk University, Brno, pp. 549–558
2. Godina M. and Matteucci P. (2003), Reductive G-structures and Lie derivatives, Journal of Geometry and Physics 47, 66–86
3. Fatibene L. and Francaviglia M. (2011), General theory of Lie derivatives for Lorentz tensors, Communications in Mathematics 19, 11–25
4. 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
• 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