In 1903, W. Boy showed that the real projective plane $\mathbb{R}P^2$ can be immersed in the Euclidean space $\mathbb{E}^3$ (see Werner Boy, Math. Ann. 57 (1903), no. 2, 151-184.). Suppose a Riemannian surface ($\mathbb{R}P^2$, g) has positive sectional curvature $K(x)$ everywhere. $K(x)$ needs not to be a constant. The question is: can this surface isometrically immersed in $\mathbb{E}^3$. I guess that there is no such an immersion. But I did not find a relative reference or a direct proof by myself.

As is well known that D. Hilbert proved that a hyperbolic plane can not be isometrically immersed in $\mathbb{E}^3$ (see D. Hilbert, Trans. Amer. Math. Soc. 2 (1901), no. 1, 87-99.). This is the motivation of the above problem.

embeddedin $\mathbb{R}^4$ smoothly, and it can be embedded in $\mathbb{R}^5$ with a metric of positive curvature, in fact with constant positive curvature.) $\endgroup$ – Robert Bryant Jan 29 at 9:42