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

2005/01/31 by C. A. Rossi, Rossi, C. A.
Mathematics · Physics and Astronomy · #57R40 #57R42 #57R56 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP #msc:57R40 #msc:57R42 #msc:57R56

paper · pdf · doi:10.48550/arxiv.math/0501546

9 pages

arxiv created 2005/01/31 · arxiv updated 2009/12/01

Abstract

In this short note, we investigate some features of the space \Injectdm 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 \Injectdm. 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 \citeBT for the 3-dimensional results and \citeCR1, \citeC for a first glimpse into higher-dimensional knot invariants).

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