Let $M$ be the manifold of all matrices in $M\_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.



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> Does $PM$ satisfy fixed point property?
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Fixed point property for the projectivization of manifold of fixed rank matrices
Usually no (check the precise dimensions yourself). In fact, just the map $M\mapsto GM$ for $G\in GL(n)$ typically has no fixed points. For $GM=\lambda M$ implies that the columns of $M$ are eigenvectors of $G$ (with the same eigenvalue $\lambda$). Thus, it suffices to assume that $G$ has no eigenspace of dimension $\ge k$.


