arXiv research

We describe a method to obtain $\mathrm{SU}(3)$-structures and $\mathrm{G}_2$-structures on 6 and 7-dimensional manifolds respectively, such that its associated metric is Einstein. More concretely, we have that different classes of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$-structures, on 5 and 6-dimensional manifolds whose…

No exact G₂-structures on compact Lie group quotients.

The paper explores strong G2-structures with torsion and their geometric properties.

Geometric structures on 5-manifolds from surface group representations of G2'.

Study on Lie groups with exact G2 structures and closed eigenforms.

New compact manifolds with closed G2 structures found.

Paper analyzes $\mathrm{U}(n)$-structures and their minimal left ideals.

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze invariant $\mathrm{Spin(7)}$ structures on quasi abelian Lie algebras.

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.

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

We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold endowed with an $\mathrm{SU}(3)$-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the $\mat…

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

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $\varphi:\mathrm{Cl}^0(E) …

These notes give an informal and leisurely introduction to $\mathrm{G}_2$ geometry for beginners. A special emphasis is placed on understanding the special linear algebraic structure in $7$ dimensions that is the pointwise model for $\mathrm{G}_2$ geometry, using the octonions. The basics of $\mathrm{G}_2$-structures a…

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characterist…

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible c…

Study harmonic flow of Spin(7)-structures on compact 8-manifolds.

The paper classifies flows of SU(2)-structures on 4-manifolds.

New stratification reveals intrinsic singularity types of orbit spaces.

Goldman symplectic form and complex structure compatible on $\mathrm{SL}(3,\mathbb R)$ Hitchin component.

Mathematical construction of vertex algebra representations from integrable G2 structures.

In this paper we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on…

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…

New formulas link string bordism to integers.

Study on deformation theory of nearly G2 manifolds with obstructions.

The paper classifies topological properties of specific geometric forms on manifolds.

Classifies 7D manifolds with specific geometric properties.

Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.

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

Classifies real-analytic SL(n,R) actions on closed manifolds.

Study solutions and singularities of G2-structures flows on specific manifolds.

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

Consider a three dimensional cusped spherical $\mathrm{CR}$ manifold $M$ and suppose that the holonomy representation of $π_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{CR}$ structure on some Dehn sur…

We study geometric structures of $\mathcal{W}_4$-type in the sense of A. Gray on a Riemannian manifold. If the structure group $\mathrm{G} \subset \SO(n)$ preserves a spinor or a non-degenerate differential form, its intrinsic torsion $Γ$ is a closed 1-form (Proposition \ref{dGamma} and Theorem \ref{Fixspinor}). Using …

We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define various quantities associated to such a hypersurface using the $\mathrm{G}_2$ structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almos…

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

New complex structures found in quantum SU(3) manifold.

Let $(M,Ω)$ be a closed $8$-dimensional manifold equipped with a generically non-integrable $\mathrm{Spin}(7)$-structure $Ω$. We prove that if $\mathrm{Hom}(H^{3}(M,\mathbb{Z}), \mathbb{Z}_{2}) = 0$ then the moduli space of irreducible $\mathrm{Spin}(7)$-instantons on $(M,Ω)$ with gauge group $\mathrm{SU}(r)$, $r\geq 2…

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Mar…

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 …

Researchers classify invariant Hermitian structures on flag manifolds with parallel Bismut torsion.

The paper classifies degenerate almost complex surfaces in a nearly Kähler space.

Study new ECS structures in 5D Minkowski compactifications of M-theory.

Operads help quantify polygon spaces, proving dimensions equal.

Topology of isometric classes and flows of geometric structures

Study compares two pseudo-Kähler structures on a specific mathematical component.

We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $H\subset\mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a semidirect product of the three-dimensional continuous Heisenberg group $H_3(\mathbb{R})$ and the…