Bozhidar Z. Iliev
The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.
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We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a flat surface with flat normal bundle. This family satisfies a permutability property.
Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles
本文在拟正则紧致Sasakian流形上引入基本向量丛的拟正则性与正则性概念，建立轨形基本群半单表示与Higgs轨形丛之间的非阿贝尔Hodge对应，并将其推广到平坦丛与基本Higgs丛，同时证明了数值平坦丛刻画的Sasakian类比。
In this paper we study the limiting behaviour of the harmonic flow on flat complex vector bundle, and prove the limit must be isomorphic to the graded flat complex vector bundle associated to the Jordan-Hölder filtration.
We construct a metrical framed structure on the tensor bundle of a Riemannian manifold equipped with a Cheeger-Gromoll type metric and by restricting this structure to the tensor sphere bundle, we obtain an almost metrical paracontact structure on the tensor sphere bundle. Moreover, we show that the tensor sphere bundles endowed with the induced metric are never space forms.
We establish a criterion for the flatness of a principal circle bundle in terms of the intrinsically harmonic form problem. It states that the flatness is equivalent to the intrinsic harmonicity of a certain natural associated form.
We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformal metrics holonomic with respect to the same coordinate system.
We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.
I prove the Bloch conjecture:all secondary characteristic classes of flat bundles over complex projective varietes are torsion, except the first.
We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal bundle, which improve a result of K. Yang. Then we derived that every Willmore two sphere with flat normal bundle in $S^n$ is conformal to a minimal surface with embedded planer ends in $\mathbb{R}^3$. We also point out that for a class of Willmore tori, they have flat normal bu…
The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend this correspondence to the projectively flat bundles case. We prove that there is an equivalence of categories between the category of $ω$-semi-stable (poly-stable) Higgs bundles $(E, \overline{\partial}_{E}, ϕ)$ with $(2rc_{2}(E)-(r-1)c_{1}^{2}(E))\cdot [ω]^{n-2}=0 $ and the catego…
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
In this paper, we use the affine Hermitian-Yang-Mills flow to prove a generalized Donaldson-Uhlenbeck-Yau theorem on flat Higgs bundles over a class of non-compact affine Gauduchon manifolds.
We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.
Hermitian-Poisson metrics on projectively flat complex vector bundles over non-compact Gauduchon manifolds
本文研究非紧Gauduchon流形上的射影平坦丛，结合热流技术与连续性方法，建立了Hermitian-Poisson度量存在性与半单性之间的对应关系。
We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoint bundle of a flat bundle whose restriction to each fiber is the class of the Maurer-Cartan 3-form. Then we use this result to define an invariant of a gauge transformation of a flat bundle which describes the effect of the gauge transformation on a Chern-Simons functional.
We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line bundles it agrees with the usual notion of the degree.
We classify conformally flat Riemannian $3-$manifolds which possesses a free isometric $S^1-$action.
We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a K-homology class $η\in K_*(M)$ twisted with an almost flat bundle in terms of the image of $η$ under Lafforgue's assembly map and the almost representation associated to the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to gener…
Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.
We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
We discuss some aspects of index and secondary index theory for flat bundles with duality. This theory was first developed by J. Lott. Our main purpose in the present paper is to provide a modification with better functorial properties.
In this note we will prove that an $n$ dimensional graphic self-shrinker in $R^{n+m}$ with flat normal bundle is a linear subspace. This result is a generalization of the corresponding result of Lu Wang in codimension one case.
We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.
In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold $(X, ω)$ with the Hermitian metric $ω$ satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistablity with vanishing first and second Chern numbers, approximate Hermitian flatness and the existence of a filtration whose quotients are Hermitian flat. This gives an affirmative answer to the question proposed by Demailly, Peternell and Schneider.
Calibrated submanifolds with flat normal bundles
该研究证明欧氏空间中满足常系数微分形式校准且法丛平坦的子流形为平面，还推广到配备平行校准的黎曼流形中，满足形状算子交换条件的校准子流形是全测地的。
We generalize reduction theorems for classical connections to operators with values in $k$-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
We show that flatness of the normal bundle is preserved under the mean curvature flow in the Euclidean space and use this to generalize a classical result for hypersurfaces due to Ecker-Huisken in the case of submanifolds with arbitrary codimension.
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function whe…
We establish Hörmander-type $L^2$-estimates for the $\overline{\partial}$-operators that hold uniformly for all nontrivial flat holomorphic line bundles on compact Kähler manifolds. Our result can be regarded as a $\overline{\partial}$-version of Ueda's lemma on the operator norm of Čech coboundaries for flat line bundles and indeed recovers the original version of Ueda's lemma for compact Kähler manifolds. A partial generalisation for $(p,0)$-forms on Ricci-flat manifolds is also given.
Counterexamples to preservation of flat normal bundles under mean curvature flow
本文构造反例，证明平均曲率流下平坦法丛性质不保持，并给出平行主法向条件不保持的例子。
We survey developments arising from Milnor's 1958 paper, "On the existence of a connection with curvature zero" and his 1977 paper, "On fundamental groups of complete affinely flat manifolds".
Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …
We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This generalizes previous results of Smoczyk, Wang and Xin, and Wang for minimal graphs with flat normal bundle.
We consider natural differential operations acting on sections of tensor vector bundles. Arrising problems can be reformulated as invariant theoretical problems (the IT-reduction). We give examples of usage of the IT-reduction. In particular, on a manifold with a connection and a Poisson structure we construct the canonical quantization.
We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP}^1\times \mathbb{CP}^1$, as well as the metrics on canonical bundles over flag manifolds, recently constructed by van Coevering.
We prove that if G is a compact connected Lie group and X is a compact connected hyper-Kahler manifold, then the L^2 metric on (the smooth locus of) the moduli space of flat G-bundles on X is a hyper-Kahler metric.
We give an explicit description of all complete $G$-invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.
The present paper is devoted to some results concerning with the complete lifts of an almost complex structure and a connection in a manifold to its (0,q)-tensor bundle along the corresponding cross-section.
We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in the affine space. An explicit parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences and surfaces with f…
We give a geometric obstruction to the non-negativity of the sectional curvature in the total spaces of certain Riemannian submersions with totally geodesic fibers; applications of this obstruction to several examples are given.
A well-known result asserts that any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic. In this note, we show that this conclusion remains valid for the larger class of Einstein manifolds. As an application, when assuming that the index of relative nullity of the immersion is a positive constant we conclude that the submanifold has the structure of a generalized cylinder over a submanifold with flat normal bundle.
We extend the notion of an almost flat bundle over a closed Riemannian manifold to bundles over simplicial complexes, and prove that up to a constant factor, this notion is invariant under pullback via maps which induce isomorphisms on fundamental groups. As an application, we show that the property of having infinite K-area only depends on the image of the fundamental class under the classifying map of the universal cover.
In this paper we study numerical properties of quotients of holomorphic log-tensors.
For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.
The main purpose of the present paper is to construct Riemannian almost product structures on the $(1,1)-$tensor bundle equipped with Cheeger-Gromoll type metric over a Riemannian manifold and present some results concerning with these structures.
Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles II
该研究在微分形式层面证明了复平坦向量丛的Riemann-Roch-Grothendieck定理的实部，推进了相关局部指标理论研究。
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Chern polynomial of the tensor product. In the second formula we express the total Chern class of the tensor product as resultant of two explicit polynomials with coefficient involving Chern classes of each vector bundles. This approach leads to determinantal formulas for the total …
We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G$, such that the tangent bundle $(TM,G)$ become conformally flat.
Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that a…
Submanifolds with constant Moebius curvature and flat normal bundle
该论文分类了满足 $n \geq 5$、$2p \leq n$ 的等距浸入 $f\colon M^{n}\to \mathbb{R}^{n+p}$，这些浸入具有常 Möbius 曲率和平坦法丛。
It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.
We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS satisfy Cartan's test and are well-posed dynamical field theories. The first family includes a constant-coefficient (cc) EDS for classical Einstein vacuum relativity (q = 4). The second family is generated only by cc 2-forms, so these are integrable (but nonlinear) systems of parti…
Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.
We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.
We make evident a curvature tensor for every vector sub-bundle of an arbitrary manifold tangent bundle which reduces to the curvature tensor of an Ehresmann connection in the case of the horizontal sub-bundle of the tangent bundle to the total space of the nonlinear fiber bundle on which the connection is defined. Then the classical theorem of Frobenius would characterize the complete integrability of a vector sub-bundle of the tangent bundle by a zero curvature tensor in the sense of our definition here. A basic tool is a result about the curvature tensor of the natural lift of the vector sub…
We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / Γ$, where $G$ is a complex connected Lie group and $Γ$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_ρ$ associated to any irreducible representation $ρ: Γ\rightarrow \text{GL}(r,{\mathbb C})$. More precisely, we prove that $E_ρ$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector bundle. All the rational Chern classes of $E$ vanish, in particular, its degree is zero. …
I consider principal Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced by Bruzzo and Graña Otero. I prove that a principal Higgs bundle $\mathfrak{E}=(E,φ)$ is H-nflat is either stable or there exists a Higgs reduction of $\mathfrak{E}$ to a parabolic subgroup $P$ of $G$ such that the principal $L$-bundle $\mathfrak{E}_L$ obtained by extending the reduced Higgs bundle $\mathfrak{E}_P$ to the Levi factor $L$ is H-nflat and stable; and as consequence, $H^{*}(\mathfrak{E},\mathbb{R})$ is isomorphic to the cohomology ring of the associated graded object $\ma…
Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vector bundles on K3 surfaces in the large complex structure limit. We define a notion of slope stability, and show that if the flat connection D has regular singularities, and the Riemannian metric g has finite volume then E admits a Poisson metric with asymptotics determined by the…
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