In 1999, DeSmet, Dillen, Verstraelen and Vrancken (DDVV) proposed the following conjecture:

Suppose that M is an n-dimension Riemannian manifold immersed into the n+m dimensional space form of constant sectional curvature c. Define the scalar curvature to be

where R is the curvature tensor with respect to the pull-back metric, and {e_i} is the orthonormal basis of the immersion. Also define the normal scalar curvature to be

，

where R' is the curvarture tensor of the normal bundle, and ｛ ξ_r} is the orthonormal basis of the normal bundle. The DDVV conjecture asserts that the following inequality to be true:

，

where H is the mean curvature tensor of the immersion.

In matrix notations, the above conjecture is equivalent to the following inequality

,

where {A_r} are symmetric matrices, and the norm of a matrix A is defined as

.

The conjecture was proved in [2].

A related conjecture is the the following Bottcher-Wenzel Conjecture [3]:

Let X,Y be two n by n matrices, then we have

||[X,Y]||² ≤2||X||²||Y||² .

References：

[1] P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. (Brno), 35(2):115-128, 1999.

[2] Z. Lu, Proof of the normal scalar curvature conjecture, preprint 2007.

[3] A. Bottcher and D. Wenzel, How big can the commutator of two matrices be and how big is it typically? Linear Algebra Appl., 403: 216-228, 2005.

[4] Z. Lu, How big is the commutator of two matrices, preprint 2007.