  Let (M,g) be a compact Riemannian manifold of dimension 3, and let
\mathscr{F} denote the collection of all embedded surfaces homeomorphic to
\mathbb{RP}^2. We study the infimum of the areas of all surfaces in
\mathscr{F}. This quantity is related to the systole of (M,g). It makes sense
whenever \mathscr{F} is non-empty.
  In this paper, we give an upper bound for this quantity in terms of the
minimum of the scalar curvature of (M,g). Moreover, we show that equality holds
if and only if (M,g) is isometric to \mathbb{RP}^3 up to scaling.
  %The proof uses the formula for the second variation of area, and Hamilton's
Ricci flow.
