Taylor Formulas for H-structures

Abstract

We give a uniform second-order Taylor formula for tensorial Riemannian $H$-structures in Riemannian normal coordinates whose initial frame is $H$-adapted. For a defining multi-tensor $\xi$ with intrinsic torsion $T$, the linear coefficient is $T_q\diamond \xi$, and the quadratic coefficient features the covariant derivative of $T$, its iterated diamond action, and curvature. The formula applies componentwise to tensors of arbitrary type and recovers the known $G_2$ and $\mathrm{Spin}(7)$ expansions. We also give the almost Hermitian specialisation and prove that vanishing of the complete second-order torsion contribution forces the intrinsic torsion to vanish at the centre of the coordinates.