arXiv research

Develops an alternative approach to Tanaka's prolongation of geometric structures.

We present Tanaka's prolongation procedure for filtered structures on manifolds discovered in [Tanaka N., J. Math. Kyoto. Univ. 10 (1970), 1-82] in a spirit of Singer-Sternberg's description of the prolongation of usual G-structures [Singer I.M., Sternberg S., J. Analyse Math. 15 (1965), 1-114; Sternberg S., Prentice-H…

Study on surface geometry in Lie groups with CR structures.

Extends Tanaka theory to supergeometry for upper bounds on supersymmetry.

We find necessary and sufficient conditions for the bi-Legendrian connection $\nabla$ associated to a bi-Legendrian structure $(\cal F,\cal G)$ on a contact metric manifold $(M,φ,ξ,η,g)$ being a metric connection and then we give conditions ensuring that $\nabla$ coincides with the (generalized) Tanaka-Webster connecti…

New criteria for effective prolongations of graded Lie algebras.

Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.

Study biharmonic hypersurfaces in Sasakian space form using Tanaka-Webster connection.

Study on curvature properties of N(κ)-contact metric manifolds with generalized Tanaka-Webster connection.

Develops a new method for constructing absolute parallelisms on CR structures.

Paper proves Whitney stratified spaces can be given a conically smooth structure.

Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are a…

We bound the symmetry algebra of a vector distribution, possibly equipped with an additional structure, by the corresponding Tanaka algebra. The main tool is the theory of weighted jets.

Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of…

We establish Bochner-type formulas for operators related to $CR$ automorphisms and spherical $CR$ structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.

Alternative proof and description of orientations for instanton moduli spaces.

We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to …

In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.

A short proof for a theorem about composite knots.

Generalizes pseudo-product structures with abnormal extremals.

Graph Lie algebras have infinite prolongation if they have a vertex of degree one.

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Ricci tensor is parallel with respect to the generalized Tanaka-Webster connection.

Let (V,(.,.)) be a pseudo-Euclidean vector space and S an irreducible Cl(V)-module. An extended translation algebra is a graded Lie algebra m = m_{-2}+m_{-1} = V+S with bracket given by ([s,t],v) = b(v.s,t) for some nondegenerate so(V)-invariant reflexive bilinear form b on S. An extended Poincaré structure on a manifo…

New characterizations of ruled real hypersurfaces in complex projective space found.

We propose the study of some kind of monopole equations directly associated with a contact structure. Through a rudimentary analysis about the solutions, we show that a closed contact 3-manifold with positive Tanaka-Webster curvature and vanishing torsion must be either not symplectically semifillable or having torsion…

We study the fillability (or embeddability) of $CR$ structures under the gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then it keeps having the same property after a short time. In the Appendix, we show the uniqueness o…

Study on real hypersurfaces in complex quadric with special connections and operators.

The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.

Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.

Classifies maximal symmetry models of CR dimension 1.

We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-We…

Study magnetic curves in Sasakian manifolds, classifying and parametrizing them.

The paper shows that certain bundles have unique volumes.

New proof of Kondo-Tanaka theorem using geometric measure theory.

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

Unified approach to geometric structure equivalence problem.

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) pair of its orders. Investigation of the corresponding Tanaka algebras leads to a new Lie-Backlund theorem. We prove that all flat …

The paper defines $η$-normality for contact and paracontact manifolds and explores their properties.

Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canoni…

The paper realizes 6 supergeometries for the Lie superalgebra D(2,1;a).

Characterizes symplectic and odd-symplectic Grassmannians using VMRT.

Study symplectification of rank 2 distributions and their connections.

Totally nondegenerate surfaces in CR dimension one are maximally homogeneous and standard.

The paper realizes Lie superalgebras G(3) and F(4) as symmetries of supergeometries.

New proof shows Lie algebras are rigid under certain conditions.

In this paper we study the foliated structure of a contact metric $(κ,μ)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation to the Boeckx's classification of contact metric $(κ,μ)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contac…