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Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory


In this paper we introduce the notion of tangent space \({\mathcal {T}}_{o} {\mathcal {G}}\) of a (not necessary smooth) subgroup \({\mathcal {G}}\) of the diffeomorphism group \({\mathcal {D}}i\!f\!f^{\infty }(M)\) of a compact manifold M. We prove that \({\mathcal {T}}_{o} {\mathcal {G}}\) is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite- or infinite-dimensional) Lie groups. The tangent Lie algebra \({\mathcal {T}}_{o} {\mathcal {G}}\) introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.

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The authors would like to thank the referee for the constructive comments and recommendations which contributed to improving the paper. The research of Z. Muzsnay was supported in part by the projects EFOP-3.6.1-16-2016-00022 and EFOP-3.6.2-16-2017-00015, co-financed by the European Union and the European Social Fund.

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Correspondence to Zoltán Muzsnay.

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Hubicska, B., Muzsnay, Z. Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory. J Geom Anal 30, 107–123 (2020).

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