On Curvature Tensors of Hermitian Manifolds

By Bo Yang, Fang Zheng. 2016

97 citations recorded by the source corpus.

In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a Kahler metric. We will call such metrics G-Kahler-like or Kahler-like, for lack of better terminologies. Such metrics are always balanced when the manifold is compact, so in a way they are more special than balanced metrics, which drew a lot of attention in the study of non-Kahler Calabi-Yau manifolds. In particular we derive various formulas on the difference between the Riemannian and Hermitian curvature tensors in terms of the torsion of the Hermitian connection. We believe that these formulas could lead to further applications in the study of Hermitian geometry with curvature assumptions.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/91b8ca684a0d53a2b6c37bb951977abc9a2449db

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Fang Zheng
2. Quanting Zhao
3. Xiaokui Yang
4. Qingsong Wang
5. Bo Yang
6. Valentino Tosatti
7. Lei Ni

In Astronomy and Astrophysics

1. Fang Zheng
2. Quanting Zhao
3. Xiaokui Yang
4. Qingsong Wang
5. Bo Yang

In Geometry and Topology

1. Fang Zheng
2. Quanting Zhao
3. Xiaokui Yang
4. Qingsong Wang
5. Bo Yang
6. Valentino Tosatti
7. Lei Ni

Citations in the corpus

• On real bisectional curvature for Hermitian manifolds (2016, methodology): “,k (Riikk +Rikki) aiak >0. This is an equivalent way to describe H>0. It is analogous to the definition of Bg >0, but not exactly the same (when the metric is not Ka¨hler-like in the sense of [22]). So we have proved the first part of Proposition 1.3. To see the second part of Proposition 1.3”
• Strominger connection and pluriclosed metrics (2019, methodology): “Note that here we adopted the usual notation for curvature tensor, unlike in [37] or [36], where the first two and last two positions were swapped.”
• On Bismut flat manifolds (2016, methodology): “ Xn i,j=1 Tk ijϕi ∧ ϕj = X 1≤i<j≤n 2 Tkϕ i ∧ ϕj Note that our Tk ij is only half of the components of the torsion τused in some other literature where the second sigma term is used. As observed in [41], when eis unitary, γand θ 2 take the following simple forms: (10) (θ 2)ij = Xn k=1 Tk ijϕk, γij”