关于从$\bbR^d$到$\bbR^m$的内射线性映射空间
On the space of injective linear maps from $\bbR^d$ into $\bbR^m$
摘要：
在这篇短文中，我们研究了从$\bbR^d$到$\bbR^m$的线性内射映射空间$\Inject{d}{m}$的一些特征;特别地，我们详细讨论了它与Stiefel流形$V_{m,d}$的关系，在这种情况下，它被看作是$\bbR^m$中$d$向量的正交规范系统的集合。最后，我们证明了Stiefel流形$V_{m,d}$是$\Inject{d}{m}$的变形收缩核.这一显著事实的一个可能的应用在于研究$\bbR^m$中高维（长）结的扰动不变量：事实上，上述变形收缩的存在性是证明位形空间积分{à} laBott-Taubes的消失引理的关键工具（三维结果见\cite{BT}，高维结不变量见\cite{CR1}、\cite{C}）。
英文摘要：
In this short note, we investigate some features of the space $\Inject{d}{m}$ of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the set of orthonormal systems of $d$ vectors in $\bbR^m$. Finally, we show that the Stiefel manifold $V_{m,d}$ is a deformation retract of $\Inject{d}{m}$. One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in $\bbR^m$: in fact, the existence of the aforementioned deformation retraction is the key tool for showing a vanishing lemma for configuration space integrals {à} la Bott--Taubes (see \cite{BT} for the 3-dimensional results and \cite{CR1}, \cite{C} for a first glimpse into higher-dimensional knot invariants).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
