On the space of injective linear maps from $bbR^d$ into $bbR^m$

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 {`a} 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).