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We show that the moduli space of metrics of nonnegative sectional curvatureon every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. Wealso show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type forwhich the moduli space of metrics of positive Ricci curvature has infinitelymany path components. Examples of closed manifolds with finite fundamentalgroup with these properties were known before only in dimensions $4k+3\geq 7$.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces