In 1979, P. Wintgen proved that the Gauss curvature, the squared mean curvature and the normal curvature of any surface E^4 always satisfy a particular inequality and this equality holds if and only if the ellipse of curvature of the surface is a circle. In 1999, P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken formulated the conjecture on generalized Wintgen inequality which is also known as the DDVV conjecture. Recently, the DDVV-conjecture was finally settled for submanifolds in real space forms, of arbitrary dimension and codimension, by Z. Lu and independently by J. Ge and Z. Tang.
Recently we obtained DDVV inequalities, also known as generalized Wintgen inequalities, for Lagrangian submanifolds in complex space forms and Legendrian submanifolds in Sasakian space forms, respectively. Some applications are given. Also we stated such inequalities for slant submanifolds in complex space forms and Sasakian space forms, respectively. Further developments are mentioned.
A formatted, more detailed abstract is available here.