arXiv research

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the…

We prove the existence of a one-parameter family of nearly parallel $G_2$-structures on the manifold $S^3\times \mathbb R^4$, which are mutually non isomorphic and invariant under the cohomogeneity one action of the group $SU(2)^3$. This family connects the two locally homogeneous nearly parallel $G_2$-structures which…

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

We prove that Einstein coclosed G_2-structures are nearly parallel.

Hitchin shows that half-flat SU(3)-structures on a 6-dimensional manifold M can be lifted to parallel G_{2}-structure on the product $M\times\mathbb{R}$. We show that Hitchin's approach can also be used to construct nearly parallel G_{2}-structures by lifting so-called nearly half-flat structures. These SU(3)-structure…

We study the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel G$_2$-structure, showing that its identity component is abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G$_2$-…

Study nearly parallel G2-structures with torus symmetry using multi-moment maps.

We study the Laplacian flow of a $\mathrm{G}_2$-structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. The…

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

Classifies 7D manifolds with specific geometric properties.

Researchers compute the ν-invariant for specific G2-structures on Aloff-Wallach spaces.

We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $Λ^3_{27…

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G_2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G_2 structure. B…

For gauge groups $U(1)$ and $SO(3)$ we classify invariant $G_2$-instantons for homogeneous coclosed $G_2$-structures on Aloff-Wallach spaces $X_{k,l}$. As a consequence, we give examples where $G_2$-instantons can be used to distinguish between different strictly nearly parallel $G_2$-structures on the same Aloff-Walla…

Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.

In some other context, the question was raised how many nearly Kähler structures exist on the sphere $§^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $λ= 12$ of the L…

Study nearly half-flat SU(3)-structures on S³×S³.

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a paralle…

We classify 7-dimensional cocalibrated $\G_2$-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection $\nabla^{\mathrm{c}}$ with totally skew-symmetric torsion and a spinor field $Ψ_1$ solving the equations in the common sector of type II superstring theory. T…

Researchers describe $G_2$-structures on $7$-sphere, finding harmonic representatives.

It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along wit…

It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-…

We study an energy functional on the universal spinor bundle over a closed $n$-dimensional spin manifold $M$. The critical points of this functional, which is modelled on the total torsion functional of $G_2$-structures in seven dimensions, are pairs of Ricci-flat metrics and real parallel spinor fields provided that $…

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional ge…

Classifies 7D manifolds with specific G2-structures and automorphisms.

We describe all almost contact metric, almost hermitian and $G_2$-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the $\nabla$-parallel spinors. In particular,…

Explicit formulas for the $G_2$-components of the Riemannian curvature tensor on a manifold with a $G_2$ structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact $G_2$ manifol…

The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact ma…

Investigates nearly Kähler and parallel G2 manifolds using Hitchin functionals.

An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N^6 can be defined as a pair of a 2-form omega and a 3-form rho. We prove that any analytic SU(3)- or SU(1,2)-structure on N^6 with d omega^2 =0 can be extended to a parallel Spin(7)- or Spin_0(3,4)-structure Phi that is defined on the trivial disc bundle N^6\t…

We find Weitzenböck formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearl…

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

The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant…

We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …

Study torsion and curvature in ACYT and AHKT 8-manifolds.

Study 3-Sasakian and G2 structures on manifolds.

Parallel spinors help characterize G2* structures and isotropic forms.

We give a brief presentation of gwistor space, which is a new concept from G_2 geometry. Then we compute the characteristic torsion T^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T^c is \nabla^c-parallel; this allows for the classif…

Half-flat SU(3)-structures are the natural initial values for Hitchin's evolution equations whose solutions define parallel G_2-structures. Together with the results of arXiv:0912.3486v1, the results of this article completely solve the existence problem of left-invariant half-flat SU(3)-structures on decomposable Lie …

Study on compact manifolds for exact G$_{2}$-Structures without additional constraints.

This is a short note on generalized $G_2$-structures obtained as a consequence of a $T$-dual construction given in a previous work of the authors together with Leonardo Soriani. Given classical $G_2$-structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized $G_2$-str…

Study on $G_2^*$ structures and almost para-contact structures in 7D.

Surveying recent progress on flows of $G_2$-structures on 7-manifolds.

We survey recent progress in the study of $G_{2}$-structure Laplacian coflows, that is, heat flows of co-closed $G_{2}$-structures. We introduce the properties of the original Laplacian coflow of $G_{2}$-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified co…

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…

Study on $G_2$-structures with negative Ricci curvature on closed and noncompact manifolds.

Study on $G_2$-structures manifold geometry.