  Let $X_{\mathbb{C}}$ be a smooth real affine variety with compact real points
$X_{\mathbb{R}}$. We show that $X_{\mathbb{C}}$ is diffeomorphic to the normal
bundle of $X_{\mathbb{R}}$ provided that $X_{\mathbb{C}}$ admits a complete
Riemannian metric of nonnegative sectional curvature which is also invariant
under the conjugation.
