arXiv research

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

Non-trivial Clifford bundle from loop space tangent bundle.

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…

Logarithmic connections on complex manifolds with trivial tangent bundle.

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

Study of symplectic trivialization and reduction of bundles with symmetry and connection.

Computes Picard groups of complex parallelizable manifolds.

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincaré equations on the matched pair of Lie algebra…

The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.

Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to be a locally trivial fiber bundle containing the tangent space to the base as a…

Study on non-Kähler Calabi-Yau manifolds and their geometric structures.

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…

This work defines a categorical notion of principal bundles.

Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson's representation of instanton bundles…

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.

In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on …

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply…

We discuss properties of the regular part $S_{reg}$ of a subcartesian space $S$. We show that $S_{reg}$ is open and dense in $S$ and the restriction to $S_{reg}$ of the tangent bundle of $S$ is locally trivial.

Let $G$ be a connected complex Lie group and $Γ\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle …

A new approach to Riemannian geometry using embedded and submersion structures.

This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an applic…

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

Study of tangent bundle positivity on complex projective varieties.

Consider a closed manifold $M$ immersed in $\R^m.$ Suppose that the trivial bundle $M\times\R^m=TM\otimes νM$ is equipped with an almost metric connection $\tilde{\nabla}$ which almost preserves the decomposition of $M\times\R^m$ into the tangent and the normal bundle. Assume moreover that the difference $Γ=\partial-\t…

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

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem…

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.

The article investigates conditions for isomorphism of singular tangent bundles.

Extends differential geometry concepts to manifolds with super tangent bundles.

We consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections…

Researchers compute the cohomology of an elliptic tangent bundle.

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

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…

Proof of orientable 3-manifolds parallelizability using knot theory.

Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.

Characterizes special curves on surface tangent bundles.

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

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…

Let $X$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $X$…

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…

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…

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…