The Schwarz genus of the Stiefel manifold and counting geometric configurations

Abstract

In this paper we compute: the Schwarz genus of the Stiefel manifold $V_k({\mathbb{R}}^n)$ with respect to the action of the Weyl group $W_k:=(\mathbb{Z}/2{)}^k⋊{\Sigma }_k$, and the Lusternik--Schnirelmann category of the quotient space $V_k({\mathbb{R}}^n)/W_k$. Furthermore, these results are used in estimating the number of: critically outscribed parallelotopes around the strictly convex body, and Birkhoff--James orthogonal bases of the normed finite dimensional vector space.

References in corpus (2)

• The genus and the category of configuration spaces
• Periodic billiard trajectories in smooth convex bodies