Principal bundle structures among second order frame bundles

Abstract

Using a model for the bundle ${\hat{\mathcal{F}}}^{2}M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal{F}}^{2}M$ of holonomic second order frames of $M$, we introduce in ${\hat{\mathcal{F}}}^{2}M$ a principal bundle structure over ${\mathcal{F}}^{2}M$, the structure group being the additive group $A_2(n)$ of skew-symmetric bilinear maps from ${\mathbb{R}}^2\times {\mathbb{R}}^n$ into ${\mathbb{R}}^n$. The composition of the projection of that structure with the existing projection of the bundle ${\tilde{\mathcal{F}}}^{2}M$ of non-holonomic second order frames of $M$ over ${\hat{\mathcal{F}}}^{2}M$ provides a principal bundle structure in ${\tilde{\mathcal{F}}}^{2}M$ over ${\mathcal{F}}^{2}M$. These results close an existing gap in the theory of second order frame bundles.