arXiv research

We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics $G$ on the space $\operatorname{Imm}(S^1,\mathbb R^2)$ of parametrized regular curves. For many metrics the tangent space $T_c\operatorname{Imm}(S^1,…

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition $[J_+,J_-] = 0$, a condition referred to as "split tangent bundle." Moreover, we show that in this in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a num…

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertic…

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

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

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

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

Newlander-Nirenberg theorem extended to complex b-manifolds.

The paper explores actions of surface mapping class groups on 3-manifolds.

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…

We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.

We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3 M$ in the situation where the tangent bundle splits under the holonomy of $\nabla$ and the torsion of $\nabla$ is of `split' type. We prove an optimal lower bound for the first eige…

This paper introduces tangent display maps to simplify tangent category theory.

The abstract proves a global splitting theorem for Poisson manifolds.

We give examples illustrating the fact that the different space/time splittings of the tangent bundle of a semi-Riemannian spin manifold give rise to non-equivalent norms on the space of compactly supported sections of the spinor bundle, and as a result, to different completions. We give a necessary and sufficient cond…

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $\nabla\colon \mathfrak{X}(M)\timesΓ(E)\toΓ(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\op…

Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.

Study non-split supermanifolds from complex manifolds.

We associate a flow $φ$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $φ$ always admits a dominated splitting and identify special cases in which $φ$ is Anosov. In particular, starting from holomorph…

Ehresmann connections in tangent categories

We study Finsler spacetimes and Killing vector fields taking care of the fact that the generalized metric tensor associated to the Lorentz-Finsler function $L$ is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vect…

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank…

A complex contact structure $γ$ is defined by a system of holomorphic local $1$-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle ${\rm Ker}\, γ$ of the tangent bundle and a line bundle $L$. In this paper, we prove that the sheaf of holomorphic $k$-vectors on a compl…

A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the cano…

Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.

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

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…

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…

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

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.

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…

Characterizes special curves on surface tangent bundles.

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

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

The paper introduces a new concept of frame vorticity and uses it to find optimal sections in specific geometric settings.

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…

Geodesics in symmetrized bidisc have intrinsic orthogonality and distinguished directions.

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…

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…

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…