arXiv research

In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.

VB-groupoids can be thought of as vector bundle objects in the category of Lie groupoids. Just as Lie algebroids are the infinitesimal counterparts of Lie groupoids, VB-algebroids correspond to the infinitesimal version of VB-groupoids. In this work we address the problem of the existence of a VB-groupoid admitting a g…

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.

Revisits Koiso's rigid metrics on complex projective spaces.

Study on complex Grassmannians' rigidity using Einstein deformations.

Study on deformation theory of nearly G2 manifolds with obstructions.

Develops deformation theory for symplectic foliations using $L_\infty$-algebras.

On pseudo-Riemannian manifolds of even dimension $n\geq 4$, with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signatur…

For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in a certain naturally defined open geodesic quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of the manifold. We show that the 1st order infinitesimal obstruction to the quadrilateral to close is…

Study on deformation cohomology for braided commutative structures.

A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the struc…

New Poisson structures defined from Lie algebroids, with conditions for existence.

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are relate…

Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show…

A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete prolongation of the Killing equation and then restrict to the curve, and 2) comput…

Study on deformations of symmetric spaces using Jordan algebras.

In this paper, we present the testing of four hypotheses on two streams of observations that are driven by Lévy processes. This is applicable for sequential decision making on the state of two-sensor systems. In one case, each sensor receives or does not receive a signal obstructed by noise. In another, each sensor rec…

Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.

Generalized differential geometry uses infinitesimals to solve singularities in differential equations.

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie …

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…

The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (Ω^*[1](\fol), d_\fol…

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally …

This paper considers the Pontryagin characters of graded vector bundles of finite rank, in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin characters vanish if the graded vector bundle carries a representation up to homotopy of the Lie algebroid. As a consequence, this gives a stron…

Develops theory of multiplicative Ehresmann connections for Lie groupoids.

In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies.…

Study of instantons on Stiefel manifold with $G_2$ and Sasakian structures.

Study of multiplicative connections in Lie groupoids.

CR 3-sphere rigidity proven through curvature invariant.

Decouples moduli groups in heterotic string theory cohomology.

A three-dimensional Riemannian manifold has locally 6, 4, 3, 2, 1 or none independent Killing vectors. We present an explicit algorithm for computing dimension of the infinitesimal isometry algebra. It branches according to the values of curvature invariants. These are relative differential invariants computed via curv…

This is a survey of our research on geometric structures of projective embeddings and includes some topics of our talks in several symposia during 1990-99. We clarify our main problem, which is to construct a kind of geometric composition series of projective embeddings. The concept of "geometric composition series" is…

Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kähler manifold $(M^{2n}, J, g)$ is symplectically dual to the Mas…

We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group $G$ to smooth maps into a homogeneous space $M=G/H$, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold $Σ\subset M…

A new framework for Einstein-Hilbert action with topological variations.

This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between…

CR Killing operator derived from tractor calculus for CR structures.

This study introduces a unified cohomology theory for braided algebras.

Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of…

Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.

We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at $\mathcal{O}(α')$ to account for the Green-Schwarz anomaly cancellation mechanism, we investigate proper…

The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with two-dimensional, two-components Poisson bracket…

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call Koszul $L_\infty$-algebra. This $L_\infty$-algebra is a cousin of the Koszul…

The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.

Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative a…

Study on bending knots and energy changes in 3D space.

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…