It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply connected manifold M which is not a sphere \((\dim M \ge 4)\) does not admit a real projective structure. Cooper and Goldman gave an example of a 3-dimensional manifold not admitting a real projective structure and this is the first known example. In this article, by generalizing their work, we construct a manifold \(M^n\) with the infinite fundamental group \({\mathbb {Z}}_2 *{\mathbb {Z}}_2\), for any \(n\ge 4\), admitting no real projective structure.

一个重要的问题是是否可以在给定的歧管上放置几何图形。众所周知，任何接纳真实射影结构的简单连接的封闭歧管都必须是球体。因此，不是球面\（（\ dim M \ ge 4）\）的任何简单连接的歧管M都不允许真实的投影结构。库珀和戈德曼举了一个3维流形的例子，它不承认真实的射影结构，这是第一个已知的例子。在这篇文章中，通过推广他们的工作，我们构建了一个歧管\（M ^ N \）与无限的基本群\（{\ mathbb {Z}} _ 2 * {\ mathbb {Z}} _ 2 \） ，对任何\ （n \ ge 4 \），不接受任何真实的投影结构。