In differential geometry, the Kosmann lift,[1][2] named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natural lift defined on the bundle of linear frames.[3]

Generalisations exist for any given reductive G-structure.

Introduction

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In general, given a subbundle   of a fiber bundle   over   and a vector field   on  , its restriction   to   is a vector field "along"   not on (i.e., tangent to)  . If one denotes by   the canonical embedding, then   is a section of the pullback bundle  , where

 

and   is the tangent bundle of the fiber bundle  . Let us assume that we are given a Kosmann decomposition of the pullback bundle  , such that

 

i.e., at each   one has   where   is a vector subspace of   and we assume   to be a vector bundle over  , called the transversal bundle of the Kosmann decomposition. It follows that the restriction   to   splits into a tangent vector field   on   and a transverse vector field   being a section of the vector bundle  

Definition

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Let   be the oriented orthonormal frame bundle of an oriented  -dimensional Riemannian manifold   with given metric  . This is a principal  -subbundle of  , the tangent frame bundle of linear frames over   with structure group  . By definition, one may say that we are given with a classical reductive  -structure. The special orthogonal group   is a reductive Lie subgroup of  . In fact, there exists a direct sum decomposition  , where   is the Lie algebra of  ,   is the Lie algebra of  , and   is the  -invariant vector subspace of symmetric matrices, i.e.   for all  

Let   be the canonical embedding.

One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle   such that

 

i.e., at each   one has     being the fiber over   of the subbundle   of  . Here,   is the vertical subbundle of   and at each   the fiber   is isomorphic to the vector space of symmetric matrices  .

From the above canonical and equivariant decomposition, it follows that the restriction   of an  -invariant vector field   on   to   splits into a  -invariant vector field   on  , called the Kosmann vector field associated with  , and a transverse vector field  .

In particular, for a generic vector field   on the base manifold  , it follows that the restriction   to   of its natural lift   onto   splits into a  -invariant vector field   on  , called the Kosmann lift of  , and a transverse vector field  .

See also

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Notes

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  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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