De Smet–Dillen–Verstraelen–Vrancken normal scalar curvature conjecture

Let $M^n$ be an immersed submanifold of a real space form with constant sectional curvature $\kappa$. The normalized scalar curvature is denoted by $\rho$, the normalized normal scalar curvature by $\rho^{\bot}$, and the normalized mean curvature vector field by $H$. DDVV conjecture.

$$\rho+\rho^{\bot}\leq |H|^2+\kappa.$$

This is the geometric normal scalar curvature conjecture underlying the DDVV-type matrix inequalities. Its resolution status is not specified in the supplied text.