发表机构
Dipartimento di Matematica “G. Peano”, Università degli studi di Torino; Department of Mathematics and Statistics, Florida International University; Universitat Internacional de Catalunya（都灵大学; 佛罗里达国际大学; 加泰罗尼亚国际大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究全斜对称Nijenhuis张量且Bismut挠率闭合的殆埃尔米特流形（almost-SKT），引入超定椭圆性条件保证受约束Bismut–Ricci流短时存在，并在六维紧流形上构造非可积例子，揭示其与广义Ricci流的联系。
AI 中文摘要
我们研究具有全斜对称的降Nijenhuis张量且其特征（Bismut）挠率闭合的殆埃尔米特流形，即almost-SKT条件。这将SKT（pluriclosed）条件推广到非可积情形。对于固定的殆复结构，这些约束关于度量是线性的。我们引入一个超定椭圆性条件，在紧流形上，该条件使得容许的无穷小形变空间为有限维，并给出受约束的Bismut–Ricci流的短时存在性。在实维数六中，almost-SKT结构满足二分法：要么殆复结构是可积的，要么斜Nijenhuis张量处处非零且具有常数范数，并且Bismut Ricci形式消失。因此，在紧致非可积的almost-SKT六流形上，受约束的Bismut–Ricci流是平稳的。在维数八中，椭圆性总是失败，而在更高维数中，Nijenhuis刚性在一个开稠密点集上成立。在任意维数中，我们刻画了降Nijenhuis张量的全斜对称性何时逐点确定相容度量（相差一个尺度因子）。在紧流形上，结合闭合挠率，这蕴含almost-SKT度量在相差尺度意义下的唯一性；受约束的Bismut–Ricci流此时是一个显式的位似解。我们还将该流与广义Ricci流联系起来，并刻画当殆复结构变化时提升演化的无穷小障碍。最后，我们在紧致六流形上构造了具有处处非零Nijenhuis张量且特征联络非平坦的非可积almost-SKT结构，包括$S^1\ imes S^2\ imes S^3$和$S^3\ imes S^3$。在这些族中，度量是固定的，相容的殆复结构构成一个$S^2$-族。这些结构具有消失的Bismut Ricci形式，因此是almost-CYT的。
英文摘要
We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut--Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut--Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut--Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including $S^1\times S^2\times S^3$ and $S^3\times S^3$. In these families, the metric is fixed and the compatible almost complex structures form an $S^2$-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.
Comments44 pages, 1 figure
