  Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are
stationary for area: the first variation of area vanishes. In this paper we
focus on surfaces of the topological type of the real projective plane $\R
P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest
area, among those mappings which are not homotopic to a constant mapping, is an
immersion. That is, $f$ is free of branch points. As a major step toward
treating minimal surfaces of the type of the projective plane, we extend the
fundamental theorem of branched immersions to the nonorientable case. We also
resolve a question on the directions of branch lines posed by Courant in 1950.
