arXiv research

The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.

The Chern-Simons forms for R-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such R-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for R-linear connections of Lie algebroids.

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise compari…

Defines Chern-Robinson connection on Robinson manifolds.

The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.

New invariant from non-acyclic flat connections.

Study on special Hermitian manifolds with specific connection properties.

Survey on metrics on non-Kähler complex manifolds.

This work extends Chern correspondence to higher gauge theory.

Geometrically constructs dilogarithm from Chern-Simons theory.

In this note, we give the correct statements of [2,Proposition 3.3 and Theorem 3.4] and a formula of the Chern curvature in terms of the curvature tensor $R^V$ of the affine connection $\nabla^V$ and the Chern tensor $P$.

From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.

Compact Kähler manifolds with positive curvature are projective and rationally connected.

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with Levi-Civita connection replaced by Chern connection. We then show that the tensor product of canonical and the anticanonical spinor bundles, called V-…

Study deforms Hermitian metrics with positive curvature.

In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles …

In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.

We define a symmetry for a Finsler space with Chern connection and investigate its implementation and properties and find a relation between them and flag curvature.

Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harm…

The paper explores flat extensions of connections and their relation to Chern-Simons invariants.

Constructs Chern-Weil classes for Cartan geometries.

We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$.…

Develops connections and Chern-Weil theory for Lie groupoids.

The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold $M$ can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simon…

Given a 3-manifold that can be written as the double of a compression body, we compute the Chern-Simons critical values for arbitrary compact connected structure groups. We also show that the moduli space of flat connections is connected when there are no reducibles.

Paper introduces Chern minimal surfaces in Hermitian surfaces and establishes identities related to their points and bundles.

We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions…

Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…

The paper calculates the Chern-Ricci form for a twisted almost Kähler structure.

This paper introduces complex Chern-Simons bundles in families setting and proves their crystalline nature.

In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{Liu…

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.

In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler …

Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…

The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can …

We study some basic properties and examples of Hermitian metrics on complex manifolds whose traces of the curvature of the Chern connection are proportional to the metric itself.

In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …

The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.

In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any metric-compatible connection is established on Finsler manifolds.

We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links…

The aim of this note is to point out that Chern characters can be computed using curvatures o ``super-connections up to homotopy'. We also present an application to the vanishing theorem for Lie algebroids which is at the origin of new secondary classes of algebroids (Fernandes), hence, in particular, of Poisson manifo…

In the late 1980s Witten used the Chern-Simons form of a connection to construct new invariants of 3-manifolds and knots, recovering in particular the Jones invariants. Since then the associated topological quantum field theory (TQFT) has served as a key example in understanding the structure of TQFTs in general. We su…

A correspondence between three-dimensional flat connections and constant curvature four-dimensional simplices is used to give a novel quantization of geometry via complex SL(2,C) Chern-Simons theory. The resulting quantum geometrical states are hence represented by the 3d blocks of analytically continued Chern-Simons t…

Categorifies Stokes coefficients in Chern-Simons theory models.

The paper classifies Hermitian manifolds with specific connection properties.