Sur le th\'eor\`eme de F. Schur pour une vari\'et\'e presque hermitienne

TL;DR

This paper proves that almost Hermitian manifolds with certain constant sectional curvature conditions are either of constant sectional curvature or Kähler, extending Schur's theorem to almost Hermitian geometry.

Contribution

It extends Schur's theorem to almost Hermitian manifolds, establishing conditions under which they have constant sectional curvature or are Kähler.

Findings

Manifolds with pointwise constant { heta}-holomorphic sectional curvature are either of constant sectional curvature or Kähler.

Manifolds with pointwise constant antiholomorphic sectional curvature and certain curvature conditions have constant antiholomorphic sectional curvature.

Results apply to almost Hermitian manifolds of dimension ≥ 6 under specified curvature conditions.

Abstract

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant {\theta}-holomorphic sectional curvature for a number {\theta} in (0,{\pi}/2) then M is of constant sectional curvature or a K\"ahler manifold of constant holomorphic sectional curvature. Theorem 2. If M is of pointwise constant antiholomorphic sectional curvature and M is an RK-manifold (or AH3-manifold), then M is of constant antiholomorphic sectional curvature.