Conformally Flat Circle Bundles over Surfaces

Volume Rigidity of Principal Circle Bundles over the Complex Projective Space

In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.

Locally conformally symplectic reduction of the cotangent bundle

In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.

Minimal surfaces in circle bundles over Riemann surfaces

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $Σ$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $Σ$.

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler-Einstein manifolds. In special cases, we give the explicit expressions of of these metrics. These examples show that there is a compact Kähler manifold $M$ and its subvariety $N$ whose codimension is greater than 1 such that there is a complete conformally Kä…

Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $Δ^k$, $k\in…

Locally conformally symplectic bundles

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω=θ\wedgeω$. A fiber bundle with LCS fiber $(F, ω,θ)$ is called LCS if the transition maps are diffeomorphisms of $F$ preserving $ω$ (and hence $θ$). In this paper, we find conditions for the total space of an LCS fiber bundle to admit an LCS form which restricts to the LCS form of the fibers. This is done by using the coupling form introduced by Sternberg and Weinstein, \cite{gls}, in the symplectic case. The construction is related to an adapted Hamiltonian action called…

Contact deformations of closed 1-forms on Torus bundles over the circle

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact forms.

On the Geometry of Circle Bundles with Special Holonomy

We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for these spaces yielding examples for complete compact Ricci flat Lorentzian manifolds and manifolds with timelike Killing vector fields. Finally we study their holonomy and obtain in particular complete examples for Lorentzian manifolds with holonomy of so called type 4.

Eta invariants of Dirac operators on Circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present p…

Conformally Osserman manifolds and conformally complex space forms

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

Conformally Invariant Powers of Spin Operators on the Sphere

We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.

Half conformally flat gradient Ricci almost solitons

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $κ$-Einstein soliton and is realized on the cotangent bundle of an affine surface.

Locally conformally Kähler solvmanifolds: a survey

A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.

Transformations of compact locally conformally Kähler manifolds

We characterize compact locally conformally Kähler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. Then we characterize the Hopf manifolds, up to holomorphic isometry, as compact l.c.K. manifolds admitting a certain closed LCR action of $\mathbb{C}^*$.

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.

三维球面上的三维向量丛的共形协变微分对称破缺算子

Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3

本文研究了三维球面上的三维向量丛与二维球面线丛之间微分对称破缺算子的构造与分类，确定了参数(λ, ν, m)的存在条件。

Iterated circle bundles and infranilmanifolds

We give short proofs of the following two facts: Iterated principal circle bundles are precisely the nilmanifolds. Every iterated circle bundle is almost flat, and hence diffeomorphic to an infranilmanifold.

Contact structures on principal circle bundles

We describe a necessary and sufficient condition for a principal circle bundle over an even-dimensional manifold to carry an invariant contact structure. As a corollary it is shown that all circle bundles over a given base manifold carry an invariant contact structure, only provided the trivial bundle does. In particular, all circle bundles over 4-manifolds admit invariant contact structures. We also discuss the Bourgeois construction of contact structures on odd-dimensional tori in this context, and we relate our results to recent work of Massot, Niederkrueger and Wendl on weak symplectic fil…

圆包上Fano流形的eta不变量

The eta invariant of a circle bundle on a Fano manifold

研究正线丛上圆包的spin-c狄拉克算子，计算其eta不变量并扩展了adiabatic参数的范围。

Almost conformally Einstein manifolds and obstructions

A Riemannian or pseudo-Riemannian (or conformal) structure is conformally Einstein if and only if there is a suitably generic parallel section of a certain vector bundle -- the so-called standard conformal tractor bundle. We show that this characterisation leads to a systematic approach to constructing obstructions to conformally Einstein metrics. Relaxing the requirement that the parallel tractor field be generic gives a natural generalisation of the Einstein equations.

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

The SzegÖ kernel on an orbifold circle bundle

The analysis of holomorphic sections of high powers $L^N$ of holomorphic ample line bundles $L\to M$ over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. The Tian-Yau-Zelditch's asymptotic expansion of the Szegö kernel of a circle bundle plays an important role in Kähler-Einstein geometry. We generalize the expansion for compact Kähler orbifolds with only finite isolated singularities and analyze the behavior of the expansion near the singularities.

Topological T-duality for general circle bundles

We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham cohomology and $K$-theory, namely by degree 3 integral classes and a less familiar kind of twist using real line bundles. We give some examples of T-dual non-oriented circle bundles and calculate their twisted $K$-theory.

Circle actions, central extensions and string structures

The caloron correspondence can be understood as an equivalence of categories between $G$-bundles over circle bundles and $LG \rtimes_ρS^1$-bundles where $LG$ is the group of smooth loops in $G$. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an $LG \rtimes_ρS^1$-bundle.

Isotopy Classification of Engel Structures on Circle Bundles

We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classification of Engel structures on circle bundles with characteristic line field tangent to the fibers.

Conformally invariant powers of the Laplacian -- A complete non-existence theorem

We show that on conformal manifolds of even dimension $n\geq 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian $Δ^{k}$ for $k>n/2$. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of $Δ^k$ for $1\le k\le n/2$, is sharp.

Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle

We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two.

Ancient Ricci Flow Solutions on Bundles

We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $κ$-collapsed and $κ$-noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions $\geq 7$. In dimension $7$ such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of $κ$-collapsed ancient solut…

Rectification of circles and quaternions

Consider a bundle of circles passing through 0 in 4-dimensional space. It is said to be rectifiable if there is a germ of diffeomorphism at 0 that takes all circles from our bundle to straight lines. We will give a classification of all rectifiable bundles of circles containing sufficiently many circles in general position. This result is surprisingly different from those in dimensions 2 and 3(Khovanskii and Izadi) due to a connection with the quaternionic algebra.

Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces

In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over Hirzebruch surfaces. Note that the contact structures in question are fillable. This may be contrasted with an earlier paper where we showed that there are infinitely many non-conjugate tori in the contactomorphism groups of certain overtwisted lens spaces (Contact cuts, Israel …

On the leafwise cohomology and dynamical zeta functions for fiber bundles over the circle

In this paper, we give concrete descriptions of leafwise cohomology groups and show the regularized determinant expression of the dynamical zeta function for fiber bundles over $S^{1}$. As applications, we show a functional equation and some formulas for special values of the dynamical zeta function.

圆上纤维丛的相对三剖分

Relative trisections of fiber bundles over the circle

本文研究定向四维圆上纤维丛的三剖分构造，提出从纤维的缝合 Heegaard 分裂出发，通过算法显式构造相对三剖分图，并利用图的粘合恢复闭纤维丛和旋流形的三剖分，同时给出四维开书分解的三剖分。

Cyclic conformally flat hypersurfaces revisited

In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectivel…

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle $T^* \M$ of a differentiable manifold $\M$ enjoys a canonical symplectic form $Ω^*$. If $(\M,\j,g,ω)$ is a pseudo-Kähler or para-Kähler $2n$-dimensional manifold, we prove that the tangent bundle $T\M$ also enjoys a natural pseudo-Kähler or para-Kähler structure $(\J,\G,Ω)$, where $Ω$ is the pull-back by $g$ of $Ω^*$ and $\G$ is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair $(\J,\G)$ and prove that: $\G$ is scalar-flat, is not Einstein unless $g$ is flat, has nonpositive (resp.\ …

Translation to Bundle Operators

We give explicit formulas for conformally invariant operators with leading term an $m$-th power of Laplacian on the product of spheres with the natural pseudo-Riemannian product metric for all $m$.

Eta Invariant and Conformal Cobordism

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (higher dimensional) conformally flat structures in a conformally invariant way. The eta invariant gives rise to a homomorphism from this group to the circle group, which can be highly nontrivial. It remains an interesting question of how to compute this group.

Laplacian operators and Q-curvature on conformally Einstein manifolds

A new definition of canonical conformal differential operators $P_k$ ($k=1,2,...)$, with leading term a $k^{\rm th}$ power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham-Jenne-Mason-Sparling (GJMS). It is shown that on conformally Einstein manifolds these agree with the $P_k$ operato…

大体积流形上带正标量曲率的圆丛

Circle bundles with PSC over large manifolds

本研究利用辛几何技术构造了无穷多类m≥4维宏观大体积流形的正标量曲率圆丛实例，其全空间宏观维度不超过⌈m/2⌉+1，并全维度解答了Gromov关于可扩流形上此类圆丛存在性的问题。

稳定平凡向量丛上带联络的不变量

Invariants of stably trivial vector bundles with connection

本文定义了稳定平凡向量丛上联络的Chern-Simons不变量，证明其可加性，并由此得到共形浸入的障碍。

On Conformal Powers of the Dirac Operator on Spin Manifolds

The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in terms of associated tractor bundles to compute these correction terms. Depending on the signature of the semi Riemannian manifold in question, the obtained conformal powers of the Dirac operator turn out to be formally self-adjoint with respect to the $L^2-$scalar product, or formall…

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$. In general, $u$ has only a finite degree of smoothness up to $Σ$. In this paper, we study obstruction flatness of hypersurfaces $Σ$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over Kähler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obst…

Yang-Mills Connections on Conformally Compact Manifolds

We study the moduli space of Yang--Mills connections on bundles over a conformally compact manifold $\overline{M}$. We prove that, for every Yang--Mills connection $A$ that satisfies an appropriate nondegeneracy condition, and for every small deformation $γ$ of $A_{|\partial\overline{M}}$, there is a Yang--Mills connection in the interior that extends $A_{|\partial\overline{M}}+γ$. As a corollary, we confirm an expectation of Witten mentioned in his foundational paper about holography [arXiv:hep-th/9802150].

Localization for Chern-Simons on Circle Bundles via Loop Groups

We consider Chern-Simons theory on 3-manifold $M$ that is the total space of a circle bundle over a 2d base $Σ$. We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level $k$ central extension of the loop group $LG$. The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley-Witten about the equivariant localization of the Chern-Simons path in…

The hyperholomorphic line bundle

We study the hyperholomorphic line bundle on a hyperkaehler manifold with circle action introduced by A Haydys, and in particular show how it transforms under a hyperkaehler quotient. Applications include ALE spaces and coadjoint orbits.

Index-energy estimates for Yang-Mills connections and Einstein metrics

We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.

Construction of Conformally Compact Einstein Manifolds

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kahler-Einstein manifolds. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in odd dimensions. As a by-product, we obtain some Riemannian products with vanishing Q-curvature.

Conformal geometry of the supercotangent and spinor bundles

We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) on sM. We study then the conf(M,g)-module structures induced on the space of differential operators acting on spinor densities and on its spaces of symbols (functions on sM). In the conformally flat case, we classify their conformal invariants, including the conformally odd powers…

关于具有双曲基本群的非球面流形上圆丛的Hopf问题

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

本文完全刻画了具有双曲基本群的闭可定向非球面流形上圆丛的绝对度大于一的自映射存在性，证明其等价于丛虚拟平凡，并由此验证了Hopf问题的强版本，同时构造了关于映射度单调的新数值不变量。

Conformal great circle flows on the three-sphere

We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector field $X$ with unit norm whose integral curves are geodesics of $g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $g$. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that $X$ is the Reeb vector field of the 1-form $λ$ obtained by contracting $g$ with $X$. We apply these results to the case of great circle flows on the 3-sphere with two obj…

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Lap…

Critical $\bar{\partial}$ problems in one complex dimension and some remarks on conformally invariant variational problems in two real dimensions

We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying geometric principle, Theorem \ref{theorem KM}, is classical and well known \cite[Theorem 1]{KM}; it gives necessary and sufficient conditions for a connection to induce a holomorphic structure on a vector bundle over a complex manifold. Here we explore the limits of this statemen…

Conformally flat submanifolds in spheres and integrable systems

E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and co…

共形 Kähler 结构

Conformally Kähler structures

本文利用带共形不变联络的向量丛平行截面刻画给定共形类中的 Kähler 度量，并导出度量共形于 Kähler 的障碍及高维 $D$ 型代数特殊度量的不变判别。

关于芬斯勒共形保圆变换的刚性

On the rigidity of Finslerian conformal circle-preserving transformations

研究前向或反向完备的贝尔瓦尔德或可逆芬斯勒流形，通过证明若其存在非平凡共形保圆变换且变换函数有临界点，则该流形为黎曼流形，还给出了其共形微分同胚的几种情况。