The Horrocks-Mumford bundle E is a famous stable complex vector bundle of rank 2 on 4-dimensional complex projective space. By construction, E has a natural Hermitian metric h1. On the other hand, stability implies the existence of a Hermitian-Einstein metric in E which is unique up to a positive scalar. Now the obvious question is if h1 is in fact the Hermitian-Einstein metric. In this note we...

We describe and construct here pseudo-Hermitian structures θ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from Kähler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metri...

We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...

A new iteration scheme for computing the sign of a matrix which has no pure imaginary eigenvalues is presented. Then, by applying a well-known identity in matrix functions theory, an algorithm for computing the geometric mean of two Hermitian positive definite matrices is constructed. Moreover, another efficient algorithm for this purpose is derived free from the computation of principal matrix...

We develop relativistic wave equations in the framework of the new non-hermitian PT quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of PT -symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that in particular the mass matrix be Hermitian also allows for models that have no cou...

The spectrum of the Hermitian Hamiltonian 1 2p 2 + 12m 2x2 + gx4 (g > 0), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian H = 1 2p 2 + 12m 2x2 − gx4, where the coupling constant g is real and positive, is PT -symmetric. As a consequence, the spectrum ofH is known to be real and positive as well. Here, it is shown that the...

In this note it is shown that almost Hermitian locally homogeneous manifolds are determined, up to local isometries, by an integer kH , the covariant derivatives of the curvature tensor up to order kH + 2 and the covariant derivatives of the complex structure up to the second order calculated at some point. An example of a Hermitian locally homogeneous manifold which is not locally isometric to...

Notions of numerical ranges and joint numerical ranges of octonion matrices are introduced. Various properties of hermitian octonion matrices related to eigenvalues and convex cones, such as the convex cone of positive semidefinite matrices, are described. As an application, convexity of joint numerical ranges of 2×2 hermitian matrices is characterized. Another application involves existence of...