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Understanding the algebraic structure underlying a manifold with a generalaffine connection is a natural problem. In this context, A. V. Gavrilovintroduced the notion of framed Lie algebra, consisting of a Lie bracket (theusual Jacobi bracket of vector fields) and a magmatic product without anycompatibility relations between them. In this work we will show that an affineconnection with curvature and torsion always gives rise to a post-Lie algebraas well as a $D$-algebra. The notions of torsion and curvature together withGavrilov's special polynomials and double exponential are revisited in thispost-Lie algebraic framework. We unfold the relations between the post-LieMagnus expansion, the Grossman-Larson product and the $K$-map, $\alpha$-map and$\beta$-map, three particular functions introduced by Gavrilov with the aim ofunderstanding the geometric and algebraic properties of the double-exponential,which can be understood as a geometric variant of the Baker-Campbell-Hausdorffformula. We propose a partial answer to a conjecture by Gavrilov, by showingthat a particular class of geometrically special polynomials is generated bytorsion and curvature. This approach unlocks many possibilities for furtherresearch such as numerical integrators and rough paths on Riemannian manifolds.

Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials