On Strominger space forms (Q2117489)

This article is concerned with a Hermitian manifold \((M^n,g)\) and its three canonical connections associated with \(g\). Namely, the Levi-Civita connection \(\nabla\), the Chern connection \(\nabla^{\mathrm{c}}\) and the Strominger or Bismut connection \(\nabla^{\mathrm{s}}\). The authors give a new contribution to the understanding of the respective holomorphic sectional curvatures \(H(X)=R_{X\overline{X}X\overline{X}}/|X|^4\), \(H^{\mathrm{c}}(X)\) and \(H^{\mathrm{s}}(X)\), defined likewise with the respective curvatures of \(\nabla,\nabla^{\mathrm{c}},\nabla^{\mathrm{s}}\), where \(X\) is a non-zero \((1,0)\)-type tangent vector. For \(n>1\), these sectional curvatures are not necessarily pointwise constant. And being constant pointwise, does not imply they are constant on the manifold. Conjecturally, on the hypotheses that such \(H\) functions are a constant \(c\), the authors point towards the metric being Kähler in case \(c\neq0\) or being flat. In Theorem 1 it is proved that if \((M^2,g)\) is a compact Hermitian surface with \(H^{\mathrm{s}}(X)=f\), where \(f\) is a function on \(M^2\). Then either \(g\) is Kähler (in which case \(f\) is a constant and the surface is a complex space form), or \(M^2\) is an isosceles Hopf surface and \(g\) an admissible metric. An isosceles Hopf surface is a quotient \(\mathbb{C}^2\backslash\{0\}/\langle\phi\rangle\) with \(\phi(z_1,z_2)=(a_1z_1,a_2z_2)\) and \(0<|a_1|=|a_2|<1\). The paper describes the admissible metric even in higher dimensions.