5 problems

DDVV conjecture. The normal scalar curvature conjecture asserts that the normal scalar curvature of $M$ satisfies the DDVV inequality.

Let $f:M^{n}\rightarrow \widetilde{M}^{m}(c)$ be an isometric immersion from $M^{n}$ into a real space form of constant curvature $c$. Let $\rho$ and $\rho^{\perp}$ denote the norm…

Let $M^n$ be an $n$-dimensional manifold isometrically immersed in the space form $N^{n+m}(c)$ of constant sectional curvature $c$. Let $h$ be the second fundamental form, let…

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 sc…

Let $\Sigma\subset\mathbb S^n$ be a closed, orientable, connected embedded hypersurface, and let $H$ denote its mean curvature. For $x\in\mathbb S^n$, define … where…