The DDVV conjecture for submanifolds of real space forms

About 20 years old · traced to 2006

Let $M^n$ be an immersed submanifold of a real space form $N^{n+m}(c)$ , where $c$ is the sectional curvature of the ambient space form, and let $\rho$ , $\rho^{\bot}$ , and $H$ denote respectively the normalized scalar curvature, normalized normal scalar curvature, and mean curvature vector of $M$ .

DDVV conjecture.

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

This conjecture generalizes the Wintgen inequality for surfaces and Chen's inequality for submanifolds in real space forms. The supplied source does not state whether the conjecture has been resolved.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The DDVV conjecture for submanifolds of real space forms
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 normalized scalar curvature and normalized normal scalar curvature of $M^{n}$ , respectively. DDVV conjecture. The inequality
$\rho \leq \left\Vert H\right\Vert ^{2}-\rho ^{\perp }+c$
should hold. The conjecture extends the Wintgen inequality from surfaces to arbitrary submanifolds in real space forms; its status is not established by the supplied source context.
source: Cihan Özgür and Adara M. Blaga, “Generalized Wintgen inequalities for submanifolds of conformally flat manifolds”, arXiv:2602.08330 (2026).

References

Primary source

Jianquan Ge and Zizhou Tang, “A survey on the DDVV conjecture”, arXiv:1006.5326 (2010).

Additional references

5 papers in this index state this conjecture (2006–2010). The statement above is taken from the most recent of them; the others are arXiv:0711.3510, arXiv:0708.2921, arXiv:math/0610709, arXiv:math/0610721.

Posted 1 month ago

The DDVV conjecture for submanifolds of real space forms