arXiv research

Study calculates Poisson cohomology for scalar multi-D brackets, finding non-trivial deformation theory.

First non-trivial examples of deformed G_2-instantons, distinguishing nearly parallel G_2-structures.

We present the extended Kuranishi space for Kodaira surface as a non-trivial example to Kontsevich and Barannikov's extended deformation theory. We provide a non-trivial example of Hertling-Manin's weak Frobenius manifold. In addition, we find that Kodaira surface is its own mirror image. Our computation is done in the…

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the …

The paper studies deformations of Poisson brackets in two dimensions, finding non-trivial cohomology groups.

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

The paper studies deformations of mixed type surfaces in Lorentz-Minkowski space.

Study on characteristic classes for foliation deformations.

The paper computes KV cochain differentials and their geometric implications.

First non-trivial examples of deformed Spin(7)-instantons constructed.

Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …

Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which ha…

Study on deformations of LC Spin(7) instantons simplifies the problem.

CAT(0) spaces can be deformed in various ways.

The study finds and disproves the existence of specific metrics on 4-manifolds.

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…

In this paper we study geodesic mappings of $n$-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such $n$-dimensional ellipsoids admit non tri…

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-triv…

We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations pr…

In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…

We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on this algebra non-equivalent to the standard Moyal product.

New method classifies hypersurfaces that can bend infinitesimally.

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescali…

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this …

Paper discusses conditions for deforming coupled Kähler-Einstein metrics.

The number of Klein-bottle leaves in taut foliations is invariant under smooth deformations.

Completes the classification of Moebius deformable hypersurfaces for dimensions 5 and above.

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…

Study of $G_2$ compactifications in M- / string theory.

Non-compact G_2 holonomy metrics that arise from a T^2 bundle over a hyper-Kahler space are discussed. These are one parameter deformations of the metrics studied by Gibbons, Lu, Pope and Stelle in hep-th/0108191. Seven-dimensional spaces with G_2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational i…

We introduce K-deformations of generalized complex structures on a compact Kahler manifold $M=(X, J)$ with an effective anti-canonical divisor and show that obstructions to K-deformations of generalized complex structures on $M$ always vanish. Applying the stability theorem of generalized Kahler structures, together wi…

Starting with a compact hyperbolic cone-manifold of dimension n > 2, we study the deformations of the metric in order to get Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial infinitesimal Einstein deforma…

New homology invariant for links in surfaces, a deformation of APS.

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sulliva…

Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the ${\rm G}_2$-structure away from s…

Starting with a compact hyperbolic cone-manifold of dimension greater than or equal to 3, we study the deformations of the metric with the aim of getting Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial i…

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Diric…

We analyse the most general N=2 supersymmetric solutions of D=11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y_7. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structur…

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general K…

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius m…

Study of hyperbolic surface laminations and their Teichmüller spaces.

We reduce the embedding problem for hypo SU(2) and SU(3)-structures to the embedding problem for hypo G2-structures into parallel Spin(7)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial G2-structure and allows us to prove that the ev…

The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.

Deformation theory for holomorphic Cartan geometries studied.

The study examines deformations of Ricci-flat ALF spaces, showing they must be Hermitian.

Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…

Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra $A_θ$ of the noncommutative torus. We show that such $A_θ$-modules have a natural interpretatio…