Contents

Kosmann lift

Introduction[edit]

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., tangentto) ${\displaystyle Q}$. 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 subspaceof${\displaystyle T_{q}E\,}$ 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[edit]

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 liftof${\displaystyle X\,}$, and a transverse vector field ${\displaystyle X_{G}\,}$.

See also[edit]

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

Notes[edit]

References[edit]

• 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