Recent developments in Wintgen inequality and Wintgen ideal submanifolds

P. Wintgen proved in [Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979)] that the Gauss curvature G and the normal curvature KD of a surface in the Euclidean 4-space E⁴ satisfy G+|KD|≤ H ², where H ² is the squared mean curvature. A surface M² in E⁴ is called a {Wintgen ideal} surface if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in E⁴ form an important family of surfaces; namely, surfaces with circular ellipse of curvature. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms, which was well-known as the DDVV conjecture. Later, the DDVV conjecture was proven by Z. Lu and by Ge and Z. Tang independently. In this paper, we provide a comprehensive survey on recent developments in Wintgen inequality and Wintgen ideal submanifolds.