On the altered holomorphic curvatures of Hermitian manifolds

Abstract

We give a systematic treatment of the growing number of curvatures of a Hermitian metric. Natural "altered" variants are introduced. Particular focus is placed on the holomorphic sectional curvature, the real bisectional curvature, and the quadratic orthogonal bisectional curvature. We show that it is necessary to consider certain variants associated with cones in R n \{0}. We exhibit the first examples that illustrate both frame dependence and frame independence of these curvatures.

Remarks on the quadratic orthogonal bisectional curvature

“…The purpose of the present short note is to describe the link between combinatorics, distance geometry, and the QOBC. These results are dispersed throughout the papers [6,7,9], since the considerations here led to solutions of problems in the fields of combinatorics and distance geometry. The papers [6,7,9] are intended for those communities, however, and not differential geometers.…”

“…In contrast with the orthogonal bisectional curvature, the QOBC is strictly weaker than the bisectional curvature, with an explicit example constructed in [18]. The QOBC has been the subject of large interest in recent years (see, e.g., [6][7][8][9][10][11]17,18,21,22,29]).…”

“…These results are dispersed throughout the papers [7,6,9], since the considerations here led to solutions of problems in the fields of combinatorics and distance geometry. The papers [7,6,9] are intended for those communities, however, and not differential geometers. The present paper is intended for this original audience.…”

“…In contrast with the orthogonal bisectional curvature, the QOBC is strictly weaker than the bisectional curvature, with an explicit example constructed in [18]. The QOBC has been the subject of large interest in recent years (see, e.g., [6,7,8,9,10,11,17,18,21,22,29]). The purpose of the present short note is to describe the link between combinatorics, distance geometry, and the QOBC.…”