arXiv research

A short proof for a theorem about composite knots.

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.

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

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statemen…

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 …

We determine the harmonic volumes for all the hyperelliptic curves. This gives a geometric interpretation of a theorem established by A. Tanaka.

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…

Eisenhart's theorem extended to sub-Riemannian metrics on specific Lie algebras.

The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.

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

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differ…

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

Paper extends theorem on covering spaces and Jordan curves.

For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizi…

Extends Tanaka theory to supergeometry for upper bounds on supersymmetry.

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are establi…

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 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…

The paper extends Liouville theorems to sub-Riemannian manifolds.

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

Study on surface geometry in Lie groups with CR structures.

In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries,…

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.

In this paper, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1,1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR manifold with nonnegative pseudohermitian bisectional c…

L. Capogna and M. Cowling showed that if $φ$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $φ$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $φ$ is in fact $C^\inft…

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

Researchers found counterexamples to a 2-jet determination theorem in higher codimension.

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

Classifies maximal symmetry models of CR dimension 1.

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not adm…

In this paper, we investigate the geometry and classification of three-dimensional CR Yamabe solitons. In the compact case, we show that any 3-dimensional CR Yamabe soliton must have constant Tanaka-Webster scalar curvature; we also obtain a classification under the assumption that their potential functions are in the …

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

Study symplectification of rank 2 distributions and their connections.

We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the ad…

New proof shows Lie algebras are rigid under certain conditions.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

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 …

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of gene…

The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a b…

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…

Study on contact forms with constant curvature on CR manifolds.

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

Alternative proof and description of orientations for instanton moduli spaces.

We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassm…

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.