Simeon Zamkovoy
We study the geometry of PQKT-connections. We find conditions to the existence of a PQKT-connection and prove that if it exists it is unique. We show that PQKT geometry persist in a conformal class of metrics.
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In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic Kähler non locally hyper paraKähler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.
In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kaehler manifolds.
We study manifolds endowed with mixed metric 3--contact structures, proving that the distribution spanned by the Reeb vector fields is integrable, with totally geodesic integral manifolds, of constant sectional curvature $k=\pm1$. We also prove a result of projectability of such structures onto paraquaternionic Kählerian structures.
In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2,1)-bundle over a paraquaternionic Kähler manifold.
Conformal paraquaternionic contact curvature and the local flatness theorem
该研究在仿四元数切触流形上定义了仿四元数切触共形曲率张量，并证明该张量为零是流形与标准平坦仿四元数切触结构局部共形等价的充要条件。
Geometry of paraquaternionic contact structures
本文引入准四元数接触结构作为para 3-Sasakian几何的推广，构造保持该结构的特殊线性联络并给出挠率与结构方程，进而定义pqc-Einstein流形，证明para 3-Sasakian空间恰为pqc-Einstein情形，并以准四元数Heisenberg群作为平坦模型。
There is considered a connection with skew symmetric torsion on a quasi-Kähler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be Kählerian. In the case when this tensor is Kählerian, some relations are obtained between its scalar curvature and the scalar curvature of other curvature tensors. Conditions are given for the considered manifolds to be isotropic-Kähler.
Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics $\{ω_t\}_t$ which equals a fixed SKT metric $ω$ for $t=0$, along a differentiable family of complex manifolds $\{M_t\}_t$.
On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex submanifold, we prove that a complex orbifold endowed with a strong KT metric admits a strong KT resolution. In this way we obtain new examples of compact simply-connected strong KT manifolds.
In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not Kähler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when $n=2$, the SKL condition is equivalent …
Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds
研究几何 Bott-Chern 形式性与 SKT 度量及 Kähler 可解流形的关系，证明 6 维幂零流形存在 SKT 度量等价于几何 Bott-Chern 形式性，并构造非零 ABC-Massey 积。
The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n).Sp(1), QKT-connection. We study the geometry of QKT-connections. We find conditions to the existence of a QKT-connection and prove that if it exists it is unique. Studying conformal transformations we obtain a lot of (compact) examples of QKT manifolds. We present a (local) description of 4-dimensional homogeneous QKT structures relying on the known result of naturally reductive homogeneous Riemannian manifolds. We con…
We study pseudoholomorphic curves in the nearly Kalher $\mathbf{CP}^3$. It is shown that a class of curves called null-torsion are in one to one correspondence with the integrals of a holomorphic contact system on the usual Kahler $\mathbb{CP}^3$ studied by Bryant. Browing Bryant's result we get plenty of such curves. Rational curves are shown to be either vertical or horizontal or null-torsion. Both horizontal and null-torsion curves can be understood by mentioned Bryant's results while vertical curves are just fibers of the twistor fibration.
We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the Kähler nullity of the almost complex structure. We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost Kähler manifolds, a class generalizing, to some algebraic extent, the …
We prove that locally any hyper-Kähler metric with torsion admits an HKT potential.
We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soulé and Fang (concerning …
In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. We use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-Kähler manifolds obtained from products of principal $S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from our de…
Strong Kähler with Torsion is the target space geometry of $(2,1)$ and $(2,0)$ supersymmetric nonlinear sigma models. We discuss how it can be represented in terms of Generalised Complex Geometry in analogy to the Gualtieri map from the geometry of $(2,2)$ supersymmetric nonlinear sigma models to Generalised Kähler Geometry.
For an almost contact metric manifold $N$, we find conditions for which either the total space of an $S^1$-bundle over $N$ or the Riemannian cone over $N$ admits a strong Kähler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on a hypersurface of an SKT manifold, and use such structures to construct new SKT manifolds via appropriate evolution equations. Hyper-Kähler with torsion (HKT) structures on the total space of an $S^1$-bundle over manifolds with three almost contact structures are also studied.
We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.
Torsion algebras of Hermitian manifolds and rigidity
该研究将埃尔米特流形的陈挠率视为全纯切丛上的斜对称复双线性乘积，证明类陈-凯勒紧致连通埃尔米特流形的挠率乘积满足雅可比恒等式，强拟正陈全纯截面曲率时挠率李代数幂零且为零，进而推出度量为凯勒度量，还构造了非单连通非有理连通Hopf曲面的正实双截曲率度量。
Hermitian connections with parallel torsion and constant holomorphic sectional curvature
本文研究具有平行挠率和常全纯截面曲率的Hermitian联络，证明非零常数迫使挠率为零，从而度量是Kähler的且局部为复空间形式，并应用于Chern和Bismut联络。
Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature
本文证明了所有复维数下平衡Bismut挠平行埃尔米特流形的非零Chern全纯截面曲率情形，确认该类紧流形若Chern全纯截面曲率非零则必为凯勒流形，完善了相关猜想的证明。
The Fino-Vezzoni conjecture on balanced Bismut torsion-parallel manifolds
该研究针对带平衡Bismut挠率平行Hermite度量的紧复流形，证明若其还容许pluriclosed Hermite度量则必容许Kähler度量，推进了Fino-Vezzoni猜想的相关研究。
Analytic Torsion for Symmetric Contact Manifolds
本文回顾Rumin复形与解析接触挠率，利用表示论及Boothby-Wang构造，将对称接触流形的接触挠率计算化为调和形式与Dolbeault上同调，并推广至等变情形以确定孤立不动点处的等变接触挠率。
Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
本文证明了所有维数下平衡Bismut挠率平行流形的常全纯截面曲率猜想，并构造反例否证满秩B猜想。
Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds
本文研究了拟统计流形上的近 Hermite 和近反 Hermite 结构，给出了近复结构可积性条件，并证明了其在特定条件下成为 Kähler、anti-Kähler 及拟 Kähler-Norden 流形的充要或必要条件。
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…
Harmonic flow of quaternion-Kähler structures
本文建立 8 维流形上 $Sp(2)Sp(1)$-结构的梯度 Dirichlet 流理论，证明近单调性、小能量长时间存在性，并构造调和 QK 孤立子及显式收敛解。
Bismut torsion parallel metrics with constant holomorphic sectional curvature
本文针对紧 Hermitian 流形常数全纯截面曲率猜想，证明其对所有非平衡 BTP 流形成立，并推出对所有 Vaisman 流形成立。
Canonical Submersions in Nearly Kähler Geometry
本文研究了近Kähler几何中平行3-(α,δ)-Sasaki流形到近Kähler轨道空间的规范子浸入，并重新证明了特定近Kähler流形到四元数Kähler流形的浸入性。
Reducible Holonomy in Closed Torsion Geometries
本文探讨了具有闭合反对称扭量和可约旋量的连接，展示其可局部定义的黎曼子流形及其基底投影几何。重新阐述了非凯勒毕舍特赫尔米特爱因斯坦流形和sHKT结构的子流形结果，并研究了半单李群上的homogeneous SKT结构的旋量分解。
The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymm…
In this article, we study Hermitian manifolds whose Bismut-Strominger connection has parallel torsion tensor, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a necessary and sufficient condition characterizing this class in terms of the curvature tensor. In particular, Bismut flat or Bismut Kähler-like manifolds are {\em BTP} manifolds, known by our earlier results. We also obtain structural results for non-balanced {\em BTP} manifolds, and classification theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP} threefolds.
In 1984, Gauduchon considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.
For any triple $(M^n, g, \nabla)$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $Ω$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated $\mathrm{G}_2$-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of $\nabla$-parallel spinors.
We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure. With an extra almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM, which is quaternionic Kahler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM. Finally a proper S^3-bundle of complex structures is introduced, expanding to TM the well known twistor bundle of M.
The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in $\SP(n)$. We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.
Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
本文证明复维数4的所有平衡Bismut挠率平行流形满足非凯勒几何中关于具有常全纯截面曲率的紧埃尔米特流形的古老猜想，为4维平衡BTP流形结构研究提供线索。
On balanced Hermitian threefolds with parallel Bismut torsion
本文刻画了所有紧致平衡且具有平行 Bismut 挠率的 Hermitian 三维流形，延续了对 BTP 流形的研究。
The parallel Bismut torsion condition and the isotropy representation of flag manifolds
本文分类了大多数旗流形上具有平行Bismut挠率的不变Hermite结构，证明在简单李群且非例外群时，该结构等价于Kähler度量或Killing/标准度量的倍数。
${\mathrm G}_2$-structures with parallel skew-symmetric torsion
本研究完成了承载和乐含于$\rm G_2$的平行反对称挠率度量联络的7维黎曼流形分类，拓展了Friedrich的上余校准情形分类，同时附带得到对应$\rm SU(3)$几何的分类列表。
Chern flat manifolds that are torsion-critical
本文研究紧陈平坦流形上的挠率临界埃尔米特度量，证明陈平坦度量为挠率临界当且仅当对应的复李群是半单的，完善了挠率临界度量的结构刻画。
In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kahler with torsion (KT) and strong hyper-Kahler with torsion (HKT) manifolds.
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The…
Sasaki with torsion manifolds and string backgrounds
本文受埃尔米特几何中Bismut的启发，研究带挠率的Sasaki流形，引入∇-爱因斯坦流形概念，推导带挠率Sasaki结构的流并证明其与广义里奇流规范等价。
Strominger connection and pluriclosed metrics
本文证明了紧Hermitian流形的Strominger联络若满足Kähler-like条件则度量必为pluriclosed，并进一步证明该条件等价于度量pluriclosed性与挠率平行性之和。
We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kahler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.
Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture
本文证明 Fino--Vezzoni 猜想在具有平行挠率的 Hermitian 联络的紧复流形上成立，即平衡度量与 pluriclosed 度量共存时必存在平行的 Kähler 度量。
This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.
We find sufficient conditions for a principal toric bundle over compact Kähler manifolds to admit Calabi-Yau connections with torsion. With the aids of a topological classification, we construct such geometry on $n(S^2\times S^4)#(n+1)(S^3\times S^3)$
Let $(M,\nabla,\langle\;,\;\rangle)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $\langle\;,\;\rangle$ and $(T^kM)_{k\geq1}$ the sequence of tangent bundles given by $T^kM=T(T^{k-1}M)$ and $T^1M=TM$. We show that, for any $k\geq1$, $T^kM$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection $\nabla^k$ and when $M$ is a Lie group and $(\nabla,\langle\;,\;\rangle)$ are left invariant there is a Lie group structure on each $T^kM$ such that $(J_k,g_k,\nabla^k)$ are left invariant. It is well-known that $(TM,J_1,g_1)$ is Kähler if…
Spin(7) metrics from Kähler Geometry
本文利用 Kähler 商与 Kähler 约化，将 Spin(7) 度量构造归结为低维平凡典范丛流形上的 PDE，并由此得到无穷多个新的显式 Spin(7) 和乐度量。
A note on compact homogeneous manifolds with Bismut parallel torsion
本文研究了具有Bismut平行torsion的紧致齐性manifold，刻画了Chern平坦BTP manifold，并证明了某些flag manifold的BTP性质与Kähler度量相关，同时提出关于Bismut连接平行曲率的猜想。
Compact homogeneous complex manifolds
该论文研究C-空间的复与埃尔米特几何，给出三类上同调结果，证明几类情形的分类结果，还得到平衡与带挠率卡拉比-丘度量的存在性结果。
We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry for $(4,1)$. We provide a generalised complex geometry formulation of hermitian geometry, generalising Gualtieri's formulation of the $(2,2)$ case. Our formulation involves a chiral version of generalised complex structure and we provide explicit formulae for the map to generalise…
We study Hermitian metrics whose Bismut connection $\nabla^B$ satisfies the first Bianchi identity in relation to the SKT condition and the parallelism of the torsion of the Bimut connection. We obtain a characterization of complex surfaces admitting Hermitian metrics whose Bismut connection satisfy the first Bianchi identity and the condition $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$, for every tangent vectors $x,y,z,w$, in terms of Vaisman metrics. These conditions, also called Bismut Kähler-like, have been recently studied in [D. Angella, A. Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with…
We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup U*(2m), a real form of GL(2m,C), otherwise denoted by SL(m,H).U(1). We show in particular how examples may be generated from quaternionic Kähler or hyperkähler manifolds with a circle action.
Geometry and symmetries of Hermitian-Einstein and instanton connection moduli spaces
本文研究Kähler带挠率流形上Hermitian-Einstein联络模空间及瞬子联络模空间的几何结构，证明特殊向量场诱导保持度量和复结构的作用，并将模空间几何建模为全环面主丛，同时探讨相关二维sigma模型的对称代数。
One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $δ$ such that $dδ$-lemma holds. We first show that such derivation $δ$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $δ$ (which is a natural, essentially necessary condition to get a $dδ$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $dδ$-lemma holds: we consider the cases when $R$ is self adjoint with respect to …
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