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Kosmann lift

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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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