Some remarks on projective Stiefel manifolds, immersions of projective spaces and spheres

Abstract:

Let ${M^m}$ be a closed smooth manifold, $M[unk]{{\mathbf {R}}^{2m}}$ an immersion and ${\tilde M^m}{ \downarrow ^\pi }{M^m}$ a double covering. For m odd we show that the normal bundle $\tilde \nu \downarrow \tilde M$ of the immersion $\tilde M{ \downarrow ^\pi }M[unk]{{\mathbf {R}}^{2m}}$ is independent of $\varphi$ and applying this to $M = {\mathbf {R}}P(m)$ reobtain the result of E. H. Brown that a symmetric immersion ${S^m}[unk]{{\mathbf {R}}^{2m}}$ is regularly homotopic to an embedding iff $m = {2^p} - 1$.