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Kosmann lift

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

Generalisations exist for any given reductive G-structure.

Introduction

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

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

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

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

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

Definition

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

Let i F S O ( M ) : F S O ( M ) F M {\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 i F S O ( M ) ( T F M ) F S O ( M ) {\displaystyle i_{\mathrm {F} _{SO}(M)}^{\ast }(T\mathrm {F} M)\to \mathrm {F} _{SO}(M)} such that

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

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

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

See also

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

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