arXiv research

We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure $\varphi$ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to $\varphi$ will converge to a to…

Torsion-free connections on $G$-structures are proven for certain groups.

Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a n…

A flow is introduced to study $G_2$-structures with divergence-free torsion.

Identifies structure of $G_2$-structures generating same Riemannian metric.

The paper studies heat flow for $G_2$-structures on compact manifolds.

Spin(7) geometry linked to multisymplectic geometry.

New flow on 7-torus yields torsion-free G2 structure.

Researchers found G2-structures with zero torsion on specific Lie groups.

This paper is a review of the twistor theory of irreducible G-structures and affine connections. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections, while the situation with holonomies of…

The study connects conic connections and torsion-free principal connections on G-structures.

The Frölicher-Nijenhuis bracket helps classify $G_2$ and Spin(7) structures.

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.

Characterizes almost Abelian Lie algebras with special $H$-structures.

We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map from a moduli space of gluing data to the moduli space of torsion-free G_2-structu…

Study of quaternion-Kähler structures via harmonic flow on 8-manifolds.

New compact manifolds with closed G2 structures found.

Classifies holonomy groups for special geometric structures.

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…

The study classifies flat solvmanifolds and finds $G_2$-structures on them.

Only flat torus in seven dimensions admits $\mathrm{G}_{2(2)}$-structure.

The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex torsion-free affine connection.

Complete classification of quaternionic skew-Hermitian symmetric spaces found.

Formula derived for $G_2$-manifolds, showing moduli spaces are incomplete.

We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed $G_2$-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free $G_2$-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsi…

The paper describes projective structures with torsion using formal frames and Thomas-Whitehead connections.

New flow for G2-structures helps find torsion-free structures.

Introduction to G2 geometry for beginners.

Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo …

We introduce the notion of an EZ-structure on a group. Delta-hyperbolic groups and CAT(0)-groups have EZ-structures. We show torsion-free groups having an EZ-structure automatically have an action by homeomorphisms on a closed (high-dimensional) ball, which is well-behaved away from a "bad limit set" in the boundary of…

The paper studies fundamental groups of orbit configuration spaces and proves their torsion-freeness.

Homotopy braid groups are shown to be torsion-free.

Study shows torsion homology growth vanishes for certain free-by-cyclic groups.

Conditions for torsion-free connections with specific curvature maps are derived.

The paper studies connections in superintegrable systems, revealing geometric insights.

The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.

Projective structures are mostly rigid at the boundary but some are not.

Groups with specific properties have vanishing $\ell^2$-Betti numbers.

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

We study the existence of invariant metrics with holonomy $G_{2(2)}^* \subset SO(4,3)$ on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free $G_{2(2)}^*$-stru…

We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of a…

The paper proves unique Levi-Civita connections on noncommutative forms.

Compact non-formal $G_2$ manifold with $b_1=1$.

In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed …

That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation…

Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.

Flow solves $G_2$ system, proving existence of torsion-free metrics.

Introduces a new $G_2$-Hilbert functional in $G_2$-geometry.