标题： 曲率，可积性，以及六维球面 显示英文标题

标题： Curvature, integrability, and the six sphere

摘要： 本注释涉及流形上几乎-埃尔米特几何与黎曼几何之间的相互作用。 这些几何可以通过曲率来相互作用。 主要结果是一个障碍方程，用于几乎复结构在具有约束截面曲率的黎曼度量下的可积性。 随后得出若干几何结论，例如假设正交复结构的列维-奇维塔协变导数的范数公式。 我们的结果导致部分恢复了一个众所周知的事实，即圆球$6$-球$S^6$不是埃尔米特的。 部分证明本质上是内在的，并在推广圆球$S^6$结果的非复性方面显示出一定的前景。 显示英文摘要

摘要： This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not hermitian. The partial proof is intrinsic in nature, and shows some level of promise when it comes to generalizing the non-complexity of the round $S^6$ result in new directions.