Geometry of Hermitian manifolds

By Kefeng Liu, Xiaokui Yang. 2010

93 citations recorded by the source corpus.

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.

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

View Paper: https://www.semanticscholar.org/paper/ada7247437d86619ee811927da7b0e3990f4f8d8

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

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Ricci Curvatures on Hermitian manifolds (2014, methodology): “-Gauduchon metric and a balanced metric simultaneously. By Corollary 1.14, they must be diﬀerent Hermitian metrics. For more works on Hermitian manifolds with specials metrics, we refer the reader to [1, 2, 4, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 28, 22, 37, 40, 39, 60, 61, 42, 50, 43, 44, 45] an”
The Chern–Ricci flow on complex surfaces (2012, methodology): “-Ricci form of ω= √ −1gijdzi ∧dzj, deﬁned by Ric(ω) = − √ −1∂∂logdetg. The Chern-Ricci ﬂow makes sense for metrics which are merely Hermitian, and in any dimension (as do ﬂows proposed in for example [49, 34]). However, for the purposes of this paper, we will restrict to the case of Gauduchon metric”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, background): “m (2.1) with arbitrary quadratic torsion term Qand proved, among other results, a short-time existence theorem [43]. The ﬂow (2.1) was extended to the almost complex setting by Vezzoni [58]. Liu-Yang [28] propose studying the ﬂow (2.1) in the case of Q= 0. In [19], Gill introduced the following para”
On Curvature Tensors of Hermitian Manifolds (2016, background): “In [16], [17], and [18], Liu and Yang gave a detailed study of Hermitian manifolds and a thorough analysis on the relationship of various Ricci tensors arising from R and R.”
On Bismut flat manifolds (2016, unknown): “We refer the readers to [37], [9], [10], [11], [12], [13], [24], [41], [19], [16], [17], [18], [33], [34], [35], [36], [25], [26], [27], [39], [40], [38] and the references therein for some recent progress in this area.”
Strominger connection and pluriclosed metrics (2019, background): “We refer the readers to [30], [14], [18], [21], [10], [11], [12], [33], [28], [29], [27], [31], [32], [22], [23], [24], [13] and the references therein for more discussions on Strominger connection, pluriclosed metric, and related topics.”
On Chern-Yamabe problem (2015, background): “One motivation for considering exactly such “complex curvature scalar”, between other natural ones [12, 13], comes from the importance of the Chern Ricci curvature in non-Kähler Calabi-Yau problems (compare, for example, [21]).”
Scalar curvature on compact complex manifolds (2017, background): “This example is studied with details in [15, 16, 21, 22, 23, 26, 27].”