arXiv research

Researchers calculate Hofer-Zehnder capacity for twisted tangent bundles over surfaces.

Establishes a correspondence between two mathematical identities.

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of $β$-diffeomorphisms and $β$-transformations. It is a starting point of an alternative…

The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.

Proves geodesic connections on 2-torus without invariant tori.

Essential obstruction found for gluing $G_{2}-$instantons with singularities.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant coho…

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…

A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,φ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $φ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and…

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and p…

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Fo…

The paper describes invariant twisted Kähler-Einstein metrics on flag varieties.

In this paper we investigate bundles whose structure group is the loop group LU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle $ξ$. This is essentially a decomposition of $ξ$ as $ζ\otimes L\mathbb C$, where $ζ$ is a finite dimensional…

Study on triviality of tangent and generalized tangent bundles of manifolds.

This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of …

We introduce a new topological sigma model, whose fields are bundle maps from the tangent bundle of a 2-dimensional world-sheet to a Dirac subbundle of an exact Courant algebroid over a target manifold. It generalizes simultaneously the (twisted) Poisson sigma model as well as the G/G-WZW model. The equations of motion…

This paper studies generic properties of connections on vector bundles, solving cohomological equations and proving opaque connections.

We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such t…

Study compact Kähler manifolds with pseudo-effective tangent bundles.

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with…

The article investigates conditions for isomorphism of singular tangent bundles.

Extends differential geometry concepts to manifolds with super tangent bundles.

The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.

Study on Ricci solitons on tangent and unit tangent bundles.

The paper defines a new structure on tangent sphere bundles and characterizes their properties.

Researchers compute the cohomology of an elliptic tangent bundle.

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.

We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…

Characterizes special curves on surface tangent bundles.

Higher-order tangent bundles have geometric structures compatible with their iterated bundle structure.

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pair…

Study positive characteristic Fano 4-folds with nef tangent bundles.

Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.

We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian…

The purpose of the present work is to study the complete and horizontal lifts of the metallic structure on tangent bundles with respect to almost product structure. We also establish fundamental formulae related to integrability and horizontal lifts of metallic structures on tangent bundles. Moreover, the study reveale…

Given a closed symplectic 4-manifold $(X,ω)$, we define a twisted version of the Gromov-Taubes invariants for $(X,ω)$, where the twisting coefficients are induced by the choice of a surface bundle over $X$. Given a fibered 3-manifold $Y$, we similarly construct twisted Lefschetz zeta functions associated with surface b…

Study on stable vector bundles over Gauduchon manifolds.

Researchers create a new metric on complex projective space bundles.

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost Kählerian manifold. This is th…

The total space of the tangent bundle of a Kähler manifold admits a canonical Kähler structure. Parallel translation identifies the space ${\Bbb{T}}$ of oriented affine lines in ${\Bbb{R}}^3$ with the tangent bundle of $S^2$. Thus, the round metric on $S^2$ induces a Kähler structure on ${\Bbb{T}}$ which turns out to h…

Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which all…

A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of …

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorp…

Sprays on Frechet manifolds connect connections and tangent structures.

The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.