Curvature characterization of Hermitian manifolds with Bismut parallel torsion

Fangyang Zheng; Quanting Zhao

arxiv: 2407.10497 · v3 · pith:SEXFYEZBnew · submitted 2024-07-15 · 🧮 math.DG

Curvature characterization of Hermitian manifolds with Bismut parallel torsion

Quanting Zhao , Fangyang Zheng This is my paper

Pith reviewed 2026-05-23 23:15 UTC · model grok-4.3

classification 🧮 math.DG

keywords Hermitian manifoldsBismut connectionparallel torsionBTP manifoldscurvature characterizationthreefolds

0 comments

The pith

Bismut parallel torsion in Hermitian manifolds is equivalent to a condition on the Bismut curvature tensor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper provides a necessary and sufficient condition for a Hermitian manifold to have parallel torsion with respect to its Bismut connection, expressed solely in terms of the Bismut curvature tensor. The condition allows identification of BTP manifolds without computing the covariant derivative of the torsion directly. Readers interested in special Hermitian geometries would find this useful as it opens the door to studying these manifolds through curvature properties and leads to a classification in three dimensions for non-balanced cases.

Core claim

The authors prove that a Hermitian manifold has Bismut parallel torsion if and only if its Bismut curvature tensor satisfies a specific algebraic condition derived from the Bianchi identities. This characterization is stated in Theorem 1.1 and forms the basis for further results on examples and properties of such manifolds, including a classification for non-balanced threefolds.

What carries the argument

The Bismut curvature tensor, which alone determines whether the torsion is parallel under the Bismut connection.

If this is right

The curvature condition implies various general properties for BTP manifolds.
Non-balanced BTP threefolds admit a classification.
Examples of BTP manifolds can be verified using the curvature criterion alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This may facilitate explicit computations for candidate metrics in higher dimensions.
The result could connect to the study of other canonical connections on Hermitian manifolds.
It opens possibilities for investigating stability or rigidity properties of BTP structures.

Load-bearing premise

The manifold is assumed to be Hermitian so that the Bismut connection is well-defined.

What would settle it

A counterexample would be a Hermitian manifold satisfying the curvature condition but with non-parallel Bismut torsion.

read the original abstract

In this article, we study Hermitian manifolds whose Bismut connection has parallel torsion, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for brevity. We obtain a necessary and sufficient condition characterizing BTP manifolds in terms of Bismut curvature tensor alone (Theorem 1.1). We also present examples and discuss some general properties for BTP manifolds, as well as give a classification result for non-balanced BTP threefolds (Theorem 1.16).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean if-and-only-if curvature condition for Bismut torsion parallel Hermitian manifolds plus a 3D classification for the non-balanced case.

read the letter

The central new piece is Theorem 1.1: a necessary and sufficient condition on the Bismut curvature tensor alone that detects when the torsion is parallel. They follow this with examples, some general properties of these BTP manifolds, and a classification of non-balanced BTP threefolds in Theorem 1.16. That is the actual advance; the rest is supporting material in the usual style of Hermitian geometry papers. The characterization looks useful inside the subfield because it removes the need to check the torsion directly and works only with curvature data that is often easier to compute. The 3D result adds concrete examples that were not previously listed. No load-bearing gaps jump out from the abstract or the stress-test note. The setup stays inside standard Hermitian geometry with the usual Bianchi identities for the Bismut connection, and the claims do not appear to reduce to tautologies or prior self-citations. The proofs themselves are not visible here, so a referee would still need to verify the algebraic steps, but nothing in the framing suggests circularity or missing assumptions. This is niche work aimed at people already working with the Bismut connection on Hermitian manifolds. A reader outside that circle will not find broader applications or new techniques. It is solid enough incremental mathematics to deserve referee time rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper studies Hermitian manifolds with Bismut-parallel torsion (BTP manifolds). It establishes a necessary and sufficient condition characterizing BTP manifolds solely in terms of the Bismut curvature tensor (Theorem 1.1). The work also supplies examples, derives general properties of BTP manifolds, and classifies non-balanced BTP threefolds (Theorem 1.16).

Significance. The curvature characterization in Theorem 1.1 supplies a direct, curvature-only test for the parallel-torsion condition, which is a concrete advance within Hermitian geometry. The three-dimensional classification (Theorem 1.16) furnishes an explicit application. The arguments rest on the standard Bianchi identities and metric compatibility of the Bismut connection; no ad-hoc parameters or invented entities appear.

minor comments (3)

[§1] §1, after the statement of Theorem 1.1: the curvature condition is written in abstract index notation; an expanded component form or an explicit reference to the (3,0) and (2,1) parts of the curvature would aid verification in the examples of §3.
[Theorem 1.16] Theorem 1.16: the non-balanced assumption is used to exclude the balanced case, but the proof sketch does not indicate whether the balanced BTP threefolds are already known or require a separate argument; a one-sentence remark would clarify the scope.
Notation: the symbol for the Bismut curvature tensor is introduced without an explicit comparison to the Chern curvature; a brief sentence relating the two would help readers accustomed to the Chern connection.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No circularity: self-contained characterization theorem

full rationale

The central result (Theorem 1.1) is an if-and-only-if characterization of BTP manifolds via a condition on the Bismut curvature tensor alone, derived from the standard definitions and Bianchi identities of the Bismut connection on Hermitian manifolds. No quoted step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or ansatz smuggled from prior work by the same authors. The setup explicitly assumes the Hermitian structure and metric-compatible connection with skew torsion; the classification (Theorem 1.16) and examples are presented as consequences rather than inputs. The derivation is therefore independent of the target result and self-contained against external benchmarks of Hermitian geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of the Bismut connection on a Hermitian manifold and the usual properties of its torsion and curvature; no free parameters, ad-hoc axioms, or new entities are mentioned in the abstract.

axioms (1)

standard math Existence and uniqueness of the Bismut connection on any Hermitian manifold
Invoked implicitly when defining BTP manifolds and their curvature.

pith-pipeline@v0.9.0 · 5602 in / 1189 out tokens · 20039 ms · 2026-05-23T23:15:29.715192+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: BTP ⇔ R^b_{XYZW}=0, R^b_{X Y Z W}=R^b_{Z W XY}, ∇^b Ric(Q)=0, Ric(Q)χ_Y=0 (type-(1,0) vectors)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 1.4: BKL = BTP + pluriclosed (no parameter or cost function)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 1 internal anchor

[1]

Agricola, T

I. Agricola, T. Friedrich, On the holonomy of connections with skew-symmetric torsion , Math. Ann. 328 (2004), no. 4, 711-748. (cited on page 1)

work page 2004
[2]

Agricola, A

I. Agricola, A. Ferreira, Einstein manifolds with skew torsion , Q. J. Math. 65 (2014), no. 3, 717-741. (cited on page 1)

work page 2014
[3]

Agricola, A

I. Agricola, A. Ferreira, T. Friedrich, The classiﬁcation of naturally reductive homogeneous spac es in dimensions n ≤ 6, Diﬀer. Geom. Appl. 39 (2015), 59-92. (cited on page 1)

work page 2015
[4]

Alexandrov, T

B. Alexandrov, T. Friedrich, N. Schoemann, Almost Hermitian 6-manifolds revisited, J. Geom. Phys. 53 (2005), No. 1, 1-30. (cited on page 1)

work page 2005
[5]

Agricola, T

I. Agricola, T. Friedrich, M. Kassuba, Eigenvalue estimates for Dirac operators with parallel cha racteristic torsion , Diﬀerential Geom. Appl. 26 (2008), no. 6, 613-624. (cited on page 1)

work page 2008
[6]

Ambrose, I.M

W. Ambrose, I.M. Singer, On homogeneous Riemannian manifolds, Duke Math. J., 25 (1958), 647-669. (cited on pages 1, 2)

work page 1958
[7]

Andrada, R

A. Andrada, R. Villacampa, Bismut connection on Vaisman manifolds, Math. Zeit. 302 (2022), 1091-1126. (cited on pages 3, 4, 19)

work page 2022
[8]

Angella, A

D. Angella, A. Otiman, A note on compatibility of special Hermitian structures arXiv2306.02981. (cited on page 5)

work page arXiv
[9]

Angella, A

D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K¨ ahler-like curvature , Commun. Anal. Geom. 30 (2022), no. 5, 961-1006. (cited on page 2)

work page 2022
[10]

Belgun, On the metric structure of non-K¨ ahler complex surfaces , Math

F. Belgun, On the metric structure of non-K¨ ahler complex surfaces , Math. Ann. 317 (2000), 1-40. (cited on page 4)

work page 2000
[11]

Belgun, Normal CR structures on S3, Math

F. Belgun, Normal CR structures on S3, Math. Zeit. 244 (2003), 125-151

work page 2003
[12]

On the metric structure of some non-K\"ahler complex threefolds

F. Belgun, On the metric structure of some non-K¨ ahler complex threefo lds, arXiv: 1208.4021. (cited on pages 5, 18, 23) 26

work page internal anchor Pith review Pith/arXiv arXiv
[13]

Bismut, A local index theorem for non-K¨ ahler manifolds, Math

J.-M. Bismut, A local index theorem for non-K¨ ahler manifolds, Math. Ann. 284 (1989), no. 4, 681-699. (cited on page 1)

work page 1989
[14]

Cordero, M

L. Cordero, M. Fern´ andez, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: D olbeault cohomology, Trans. Amer. Math. Soc., 352 (2000), no. 12, 5405-5433. (cited on pages 4, 21)

work page 2000
[15]

Cordero, M

L. Cordero, M. Fern´ andez, L. Ugarte, Pseudo-K¨ ahler metrics on six-dimensional nilpotent Lie a lgebras, J. Geom. Phys. 50 (2004), 115-137. (cited on page 21)

work page 2004
[16]

Cleyton, A

R. Cleyton, A. Moroianu, U. Semmelmann, Metric connections with parallel skew-symmetric torsion , Adv. Math., 378 (2021), Paper No. 107519, 50 pp. (cited on page 1)

work page 2021
[17]

A. Fino, N. Tardini, Some remarks on Hermitian manifolds satisfying K¨ ahler-like conditions, Math. Zeit. 298 (2021), 49-68. (cited on page 2)

work page 2021
[18]

A. Fino, N. Tardini, L. Vezzoni, Pluriclosed and Strominger K¨ ahler-like metrics compatib le with abelian complex structures, Bull. London Math. Soc. 54 (2022), no. 5, 1862-1872. (cited on page 2)

work page 2022
[19]

Fino and L

A. Fino and L. Vezzoni, Special Hermitian metrics on compact solvmanifolds , J. Geom. Phys. 91 (2015), 40-53. (cited on page 5)

work page 2015
[20]

Q. Gao, Q. Zhao, F. Zheng, Maximal nilpotent complex structures , Transform. Groups, 28 (2023), pages 241-284. (cited on page 21)

work page 2023
[21]

Istrati, Vaisman manifolds with vanishing ﬁrst Chern class , arXiv:2304.02582

N. Istrati, Vaisman manifolds with vanishing ﬁrst Chern class , arXiv:2304.02582. (cited on page 18)

work page arXiv
[22]

Kamber, Ph

F. Kamber, Ph. Tondeur, Flat manifolds with parallel torison, J. Diﬀ. Geom., 2 (1968), 385-389. (cited on page 1)

work page 1968
[23]

Lafuente, J

R. Lafuente, J. Stanﬁeld, Hermitian manifolds with ﬂat Gauduchon connections , to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. (cited on page 21)

work page
[24]

Y. Li, F. Zheng, Complex nilmanifolds with constant holomorphic sectional curvature, Proc. Amer. Math. Soc., 150 (2022), 319-326. (cited on page 21)

work page 2022
[25]

L. Ni, F. Zheng, Hermitian manifolds whose Chern connection is Ambrose-Sin ger, Trans. Amer. Math. Soc. 376 (2023), no. 9, 6681-6707. (cited on page 2)

work page 2023
[26]

L. Ni, F. Zheng, A classiﬁcation of locally Chern homogeneous Hermitian man ifolds, arxiv.2301.00579. (cited on page 2)

work page arXiv
[27]

Ornea, M

L. Ornea, M. Verbitsky, Structure theorem of compact Vaisman manifolds , Math. Res. Lett. 10 (2003), 799-805. (cited on page 4)

work page 2003
[28]

Ornea, M

L. Ornea, M. Verbitsky, A Calabi-Yau theorem for Vaisman manifolds , arXiv:2206.08808. (cited on page 18)

work page arXiv
[29]

Popovici, Limits of projective manifolds under holomorphic deformat ions: Hodge numbers and strongly Gauduchon metrics, Invent

D. Popovici, Limits of projective manifolds under holomorphic deformat ions: Hodge numbers and strongly Gauduchon metrics, Invent. Math. 194 (2013), no. 3, 515-534. (cited on page 4)

work page 2013
[30]

Salamon, Complex structures on nilpotent Lie algebras , J

S. Salamon, Complex structures on nilpotent Lie algebras , J. Pure Appl. Algebra, 157 (2001), 311-333. (cited on page 21)

work page 2001
[31]

Schoemann, Almost Hermitian structures with parallel torsion , J

N. Schoemann, Almost Hermitian structures with parallel torsion , J. Geom. Phys. 57 (2007), No. 11, 2187-2212. (cited on page 1)

work page 2007
[32]

Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math

K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J., 7 (1978), 206-213. (cited on page 1)

work page 1978
[33]

Streets, G

J. Streets, G. Tian, A parabolic ﬂow of pluriclosed metrics , Int. Math. Res. Notices, 16 (2010), 3101-3133. (cited on page 5)

work page 2010
[34]

Strominger, Superstrings with Torsion, Nuclear Phys

A. Strominger, Superstrings with Torsion, Nuclear Phys. B 274 (1986), 253-284. (cited on page 1)

work page 1986
[35]

Tsukada, Holomorphic forms and holomorphic vector ﬁelds on compact g eneralized Hopf manifolds , Compositio Math

K. Tsukada, Holomorphic forms and holomorphic vector ﬁelds on compact g eneralized Hopf manifolds , Compositio Math. 93 (1994), no. 1, 1-22. (cited on page 18)

work page 1994
[36]

Tsukada, Holomorphic maps of compact generalized Hopf manifolds , Geom

K. Tsukada, Holomorphic maps of compact generalized Hopf manifolds , Geom. Dedicata 68 (1997), 67-71. (cited on page 18)

work page 1997
[37]

Ugarte, Hermitian structures on six-dimensional nilmanifolds , Transform

L. Ugarte, Hermitian structures on six-dimensional nilmanifolds , Transform. Groups, 12 (2007), no. 1, 175-202. (cited on page 21)

work page 2007
[38]

W ang, B

Q. W ang, B. Yang, F. Zheng, On Bismut ﬂat manifolds, Trans. Amer. Math. Soc. 373 (2020), 5747-5772. (cited on page 7)

work page 2020
[39]

Vezzoni, B

L. Vezzoni, B. Yang, F. Zheng, Lie groups with ﬂat Gauduchon connections , Math. Zeit. 293 (2019), 597-608. (cited on page 21)

work page 2019
[40]

B. Yang, F. Zheng, On curvature tensors of Hermitian manifolds, Comm. Anal. Geom. 26 (2018), no. 5, 1193-1220. (cited on pages 6, 7)

work page 2018
[41]

B. Yang, F. Zheng, On compact Hermitian manifolds with ﬂat Gauduchon conmnect ions, Acta Math. Sinica (English Series). 34 (2018), 1259-1268. (cited on page 6)

work page 2018
[42]

S.-T. Yau, Q. Zhao, F. Zheng, On Strominger K¨ ahler-like manifolds with degenerate torsion, Trans. Amer. Math. Soc. 376 (2023), no.5, 3063-3085. (cited on pages 2, 4, 6, 18, 19, 20, 23, 25)

work page 2023
[43]

Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics , J. Reine Angew. Math., 796 (2023), 245-267. (cited on pages 2, 4, 6, 7, 9, 10, 11, 12, 22)

work page 2023
[44]

Q. Zhao, F. Zheng, Complex nilmanifolds and K¨ ahler-like connections, J. Geom. Phys. 146 (2019). (cited on pages 4, 21)

work page 2019
[45]

Q. Zhao, F. Zheng, On Hermitian manifolds with Bismut-Strominger parallel to sion, arXiv: 2208.03071. (cited on page 6)

work page arXiv
[46]

Q. Zhao, F. Zheng, Bismut K¨ ahler-like manifolds of dimensions 4 and 5, arXiv: 2303.09267. (cited on page 2)

work page arXiv
[47]

W. Zhou, F. Zheng, Hermitian threefolds with vanishing real bisectional curv ature, Sci. China Math. (Chinese series), 2022, 52: 757-764, doi: 10.1360/SCM-2021-0109. (cited on p age 6)

work page doi:10.1360/scm-2021-0109 2022
[48]

Zheng, Complex diﬀerential geometry , AMS/IP Studies in Advanced Mathematics, 18

F. Zheng, Complex diﬀerential geometry , AMS/IP Studies in Advanced Mathematics, 18. American Math ematical Society, Providence, RI; International Press, Boston, MA, 2000. (cited on page 6)

work page 2000
[49]

Zheng, Some recent progress in non-K¨ ahler geometry, Sci

F. Zheng, Some recent progress in non-K¨ ahler geometry, Sci. China Math., 62 (2019), no.11, 2423-2434. (cited on page 6) 27 Quanting Zhao. School of Mathematics and Statistics, and Hub ei Key Laboratory of Mathematical Sci- ences, Central China Normal University, P.O. Box 71010, Wuhan 430079, P. R. China. Email address : zhaoquanting@126.com;zhaoquanting...

work page 2019

[1] [1]

Agricola, T

I. Agricola, T. Friedrich, On the holonomy of connections with skew-symmetric torsion , Math. Ann. 328 (2004), no. 4, 711-748. (cited on page 1)

work page 2004

[2] [2]

Agricola, A

I. Agricola, A. Ferreira, Einstein manifolds with skew torsion , Q. J. Math. 65 (2014), no. 3, 717-741. (cited on page 1)

work page 2014

[3] [3]

Agricola, A

I. Agricola, A. Ferreira, T. Friedrich, The classiﬁcation of naturally reductive homogeneous spac es in dimensions n ≤ 6, Diﬀer. Geom. Appl. 39 (2015), 59-92. (cited on page 1)

work page 2015

[4] [4]

Alexandrov, T

B. Alexandrov, T. Friedrich, N. Schoemann, Almost Hermitian 6-manifolds revisited, J. Geom. Phys. 53 (2005), No. 1, 1-30. (cited on page 1)

work page 2005

[5] [5]

Agricola, T

I. Agricola, T. Friedrich, M. Kassuba, Eigenvalue estimates for Dirac operators with parallel cha racteristic torsion , Diﬀerential Geom. Appl. 26 (2008), no. 6, 613-624. (cited on page 1)

work page 2008

[6] [6]

Ambrose, I.M

W. Ambrose, I.M. Singer, On homogeneous Riemannian manifolds, Duke Math. J., 25 (1958), 647-669. (cited on pages 1, 2)

work page 1958

[7] [7]

Andrada, R

A. Andrada, R. Villacampa, Bismut connection on Vaisman manifolds, Math. Zeit. 302 (2022), 1091-1126. (cited on pages 3, 4, 19)

work page 2022

[8] [8]

Angella, A

D. Angella, A. Otiman, A note on compatibility of special Hermitian structures arXiv2306.02981. (cited on page 5)

work page arXiv

[9] [9]

Angella, A

D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K¨ ahler-like curvature , Commun. Anal. Geom. 30 (2022), no. 5, 961-1006. (cited on page 2)

work page 2022

[10] [10]

Belgun, On the metric structure of non-K¨ ahler complex surfaces , Math

F. Belgun, On the metric structure of non-K¨ ahler complex surfaces , Math. Ann. 317 (2000), 1-40. (cited on page 4)

work page 2000

[11] [11]

Belgun, Normal CR structures on S3, Math

F. Belgun, Normal CR structures on S3, Math. Zeit. 244 (2003), 125-151

work page 2003

[12] [12]

On the metric structure of some non-K\"ahler complex threefolds

F. Belgun, On the metric structure of some non-K¨ ahler complex threefo lds, arXiv: 1208.4021. (cited on pages 5, 18, 23) 26

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

Bismut, A local index theorem for non-K¨ ahler manifolds, Math

J.-M. Bismut, A local index theorem for non-K¨ ahler manifolds, Math. Ann. 284 (1989), no. 4, 681-699. (cited on page 1)

work page 1989

[14] [14]

Cordero, M

L. Cordero, M. Fern´ andez, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structures: D olbeault cohomology, Trans. Amer. Math. Soc., 352 (2000), no. 12, 5405-5433. (cited on pages 4, 21)

work page 2000

[15] [15]

Cordero, M

L. Cordero, M. Fern´ andez, L. Ugarte, Pseudo-K¨ ahler metrics on six-dimensional nilpotent Lie a lgebras, J. Geom. Phys. 50 (2004), 115-137. (cited on page 21)

work page 2004

[16] [16]

Cleyton, A

R. Cleyton, A. Moroianu, U. Semmelmann, Metric connections with parallel skew-symmetric torsion , Adv. Math., 378 (2021), Paper No. 107519, 50 pp. (cited on page 1)

work page 2021

[17] [17]

A. Fino, N. Tardini, Some remarks on Hermitian manifolds satisfying K¨ ahler-like conditions, Math. Zeit. 298 (2021), 49-68. (cited on page 2)

work page 2021

[18] [18]

A. Fino, N. Tardini, L. Vezzoni, Pluriclosed and Strominger K¨ ahler-like metrics compatib le with abelian complex structures, Bull. London Math. Soc. 54 (2022), no. 5, 1862-1872. (cited on page 2)

work page 2022

[19] [19]

Fino and L

A. Fino and L. Vezzoni, Special Hermitian metrics on compact solvmanifolds , J. Geom. Phys. 91 (2015), 40-53. (cited on page 5)

work page 2015

[20] [20]

Q. Gao, Q. Zhao, F. Zheng, Maximal nilpotent complex structures , Transform. Groups, 28 (2023), pages 241-284. (cited on page 21)

work page 2023

[21] [21]

Istrati, Vaisman manifolds with vanishing ﬁrst Chern class , arXiv:2304.02582

N. Istrati, Vaisman manifolds with vanishing ﬁrst Chern class , arXiv:2304.02582. (cited on page 18)

work page arXiv

[22] [22]

Kamber, Ph

F. Kamber, Ph. Tondeur, Flat manifolds with parallel torison, J. Diﬀ. Geom., 2 (1968), 385-389. (cited on page 1)

work page 1968

[23] [23]

Lafuente, J

R. Lafuente, J. Stanﬁeld, Hermitian manifolds with ﬂat Gauduchon connections , to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. (cited on page 21)

work page

[24] [24]

Y. Li, F. Zheng, Complex nilmanifolds with constant holomorphic sectional curvature, Proc. Amer. Math. Soc., 150 (2022), 319-326. (cited on page 21)

work page 2022

[25] [25]

L. Ni, F. Zheng, Hermitian manifolds whose Chern connection is Ambrose-Sin ger, Trans. Amer. Math. Soc. 376 (2023), no. 9, 6681-6707. (cited on page 2)

work page 2023

[26] [26]

L. Ni, F. Zheng, A classiﬁcation of locally Chern homogeneous Hermitian man ifolds, arxiv.2301.00579. (cited on page 2)

work page arXiv

[27] [27]

Ornea, M

L. Ornea, M. Verbitsky, Structure theorem of compact Vaisman manifolds , Math. Res. Lett. 10 (2003), 799-805. (cited on page 4)

work page 2003

[28] [28]

Ornea, M

L. Ornea, M. Verbitsky, A Calabi-Yau theorem for Vaisman manifolds , arXiv:2206.08808. (cited on page 18)

work page arXiv

[29] [29]

Popovici, Limits of projective manifolds under holomorphic deformat ions: Hodge numbers and strongly Gauduchon metrics, Invent

D. Popovici, Limits of projective manifolds under holomorphic deformat ions: Hodge numbers and strongly Gauduchon metrics, Invent. Math. 194 (2013), no. 3, 515-534. (cited on page 4)

work page 2013

[30] [30]

Salamon, Complex structures on nilpotent Lie algebras , J

S. Salamon, Complex structures on nilpotent Lie algebras , J. Pure Appl. Algebra, 157 (2001), 311-333. (cited on page 21)

work page 2001

[31] [31]

Schoemann, Almost Hermitian structures with parallel torsion , J

N. Schoemann, Almost Hermitian structures with parallel torsion , J. Geom. Phys. 57 (2007), No. 11, 2187-2212. (cited on page 1)

work page 2007

[32] [32]

Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math

K. Sekigawa, Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J., 7 (1978), 206-213. (cited on page 1)

work page 1978

[33] [33]

Streets, G

J. Streets, G. Tian, A parabolic ﬂow of pluriclosed metrics , Int. Math. Res. Notices, 16 (2010), 3101-3133. (cited on page 5)

work page 2010

[34] [34]

Strominger, Superstrings with Torsion, Nuclear Phys

A. Strominger, Superstrings with Torsion, Nuclear Phys. B 274 (1986), 253-284. (cited on page 1)

work page 1986

[35] [35]

Tsukada, Holomorphic forms and holomorphic vector ﬁelds on compact g eneralized Hopf manifolds , Compositio Math

K. Tsukada, Holomorphic forms and holomorphic vector ﬁelds on compact g eneralized Hopf manifolds , Compositio Math. 93 (1994), no. 1, 1-22. (cited on page 18)

work page 1994

[36] [36]

Tsukada, Holomorphic maps of compact generalized Hopf manifolds , Geom

K. Tsukada, Holomorphic maps of compact generalized Hopf manifolds , Geom. Dedicata 68 (1997), 67-71. (cited on page 18)

work page 1997

[37] [37]

Ugarte, Hermitian structures on six-dimensional nilmanifolds , Transform

L. Ugarte, Hermitian structures on six-dimensional nilmanifolds , Transform. Groups, 12 (2007), no. 1, 175-202. (cited on page 21)

work page 2007

[38] [38]

W ang, B

Q. W ang, B. Yang, F. Zheng, On Bismut ﬂat manifolds, Trans. Amer. Math. Soc. 373 (2020), 5747-5772. (cited on page 7)

work page 2020

[39] [39]

Vezzoni, B

L. Vezzoni, B. Yang, F. Zheng, Lie groups with ﬂat Gauduchon connections , Math. Zeit. 293 (2019), 597-608. (cited on page 21)

work page 2019

[40] [40]

B. Yang, F. Zheng, On curvature tensors of Hermitian manifolds, Comm. Anal. Geom. 26 (2018), no. 5, 1193-1220. (cited on pages 6, 7)

work page 2018

[41] [41]

B. Yang, F. Zheng, On compact Hermitian manifolds with ﬂat Gauduchon conmnect ions, Acta Math. Sinica (English Series). 34 (2018), 1259-1268. (cited on page 6)

work page 2018

[42] [42]

S.-T. Yau, Q. Zhao, F. Zheng, On Strominger K¨ ahler-like manifolds with degenerate torsion, Trans. Amer. Math. Soc. 376 (2023), no.5, 3063-3085. (cited on pages 2, 4, 6, 18, 19, 20, 23, 25)

work page 2023

[43] [43]

Q. Zhao, F. Zheng, Strominger connection and pluriclosed metrics , J. Reine Angew. Math., 796 (2023), 245-267. (cited on pages 2, 4, 6, 7, 9, 10, 11, 12, 22)

work page 2023

[44] [44]

Q. Zhao, F. Zheng, Complex nilmanifolds and K¨ ahler-like connections, J. Geom. Phys. 146 (2019). (cited on pages 4, 21)

work page 2019

[45] [45]

Q. Zhao, F. Zheng, On Hermitian manifolds with Bismut-Strominger parallel to sion, arXiv: 2208.03071. (cited on page 6)

work page arXiv

[46] [46]

Q. Zhao, F. Zheng, Bismut K¨ ahler-like manifolds of dimensions 4 and 5, arXiv: 2303.09267. (cited on page 2)

work page arXiv

[47] [47]

W. Zhou, F. Zheng, Hermitian threefolds with vanishing real bisectional curv ature, Sci. China Math. (Chinese series), 2022, 52: 757-764, doi: 10.1360/SCM-2021-0109. (cited on p age 6)

work page doi:10.1360/scm-2021-0109 2022

[48] [48]

Zheng, Complex diﬀerential geometry , AMS/IP Studies in Advanced Mathematics, 18

F. Zheng, Complex diﬀerential geometry , AMS/IP Studies in Advanced Mathematics, 18. American Math ematical Society, Providence, RI; International Press, Boston, MA, 2000. (cited on page 6)

work page 2000

[49] [49]

Zheng, Some recent progress in non-K¨ ahler geometry, Sci

F. Zheng, Some recent progress in non-K¨ ahler geometry, Sci. China Math., 62 (2019), no.11, 2423-2434. (cited on page 6) 27 Quanting Zhao. School of Mathematics and Statistics, and Hub ei Key Laboratory of Mathematical Sci- ences, Central China Normal University, P.O. Box 71010, Wuhan 430079, P. R. China. Email address : zhaoquanting@126.com;zhaoquanting...

work page 2019