Real affine varieties of nonnegative curvature
  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. 
Xiaoyang Chen
math.DG