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arXiv 1504.02679math.DG
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二阶标架丛之间的主丛结构

Principal bundle structures among second order frame bundles

M. de León, A. Martín Méndez

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摘要：

利用流形$M$的二阶半完整标架丛$\hat{\mathcal F}^2M$作为$M$的二阶完整标架丛${\mathcal F}^2M$的推广，在$\hat{\mathcal F}^2M$中引入${\mathcal F}^2M$上的主丛结构，结构群为从${\Bbb R}^2 \times {\Bbb R}^n$到${\Bbb R}^n$的反对称双线性映射的加群$A_2(n)$.该结构的投影与$M$的非完整二阶标架的丛$\tilde{\mathcal F}^2M$在$\hat{\mathcal F}^2M$上的现有投影的合成提供了${\mathcal F}^2M$上的$\tilde{\mathcal F}^2M$中的主丛结构。这些结果填补了二阶标架丛理论中的一个空白。

英文摘要：

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

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