On codimension-1 submanifolds of the real and complex projective space

Inspired by the analogous result in the algebraic setting (Theorem 1) we show (Theorem 2) that the product $M\times \mathbb{R}{P}^n$ of a closed and orientable topological manifold $M$ with the $n$-dimensional real projective space cannot be topologically locally flat embedded into $$\mathbb{R}$P^{m + n + 1}$foralleven$n > m$.