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

## TL;DR

This paper proves that certain product manifolds involving real projective spaces cannot be embedded in higher-dimensional real projective spaces under specific conditions, extending algebraic results to topological settings.

## Contribution

It establishes a topological non-embedding result for products of closed orientable manifolds with real projective spaces, inspired by algebraic analogs.

## Key findings

- Product manifolds with real projective spaces cannot be locally flat embedded in certain projective spaces for even dimensions.
- The result generalizes algebraic embedding theorems to topological manifolds.
- Provides new insights into the topology of projective space embeddings.

## Abstract

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 $\mathbb{R}P^{m + n + 1}$ for all even $n > m$.