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

Abstract

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.