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A Wintgen type inequality for surfaces in 4D neutral pseudo-Riemannian space forms and its applications to minimal immersions

Let $M$ be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form $R^4_2(c)$ with constant sectional curvature $c$ and index two. In the first part of this article, we prove that the Gauss curvature $K$, the normal curvature $K^D$, and mean curvature vector $H$ of $M$ satisfy the general inequality: $K+K^D\geq H,H +c$. In the second part, we investigate space-like minimal surfaces in $R^4_2(c)$ which satisfy the equality case of the inequality identically. Several classification results in this respect are then obtained.