arXiv research

We study compactifications of type II theories on SU(2) x SU(2) structure manifolds to six, five and four spacetime dimensions. We use the framework of generalized geometry to describe the NS-NS sector of such compactifications and derive the structure of their moduli spaces. We show that in contrast to SU(3) x SU(3) s…

We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\em balanced} SU(2)-{\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-st…

Method to create Einstein SU(3) and G2 structures from specific metrics.

Study coclosed G2-structures on SU(2)²-invariant manifolds.

Invariant description of SU(2)-structures on 5-manifolds developed.

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

The paper explores SU(4)-holonomy via Hitchin flow and hypo SU(3)-structures.

Construct $SU(3)$ structures on bundles over complex manifolds.

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

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G_2-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G_2-structure with the shape tensor of the hypersurface. Additionaly, we show how to compute…

Harmonicity conditions for $SU(3)$ and $G_2$ structures derived from spinors.

We generalize Calabi-Yau 3-folds from the special Lagrangian perspective. More precisely, we study SU(3)-structures which admit as "nice" a local special Lagrangian geometry as the flat $\mathbf{C}^3$ or a Calabi-Yau structure does. The underlying almost complex structure may not be integrable. Such SU(3)-structures ar…

We review coupled ${\rm SU}(3)$-structures, also known in the literature as restricted half-flat structures, in relation to supersymmetry. In particular, we study special classes of examples admitting such structures and the behaviour of flows of ${\rm SU}(3)$-structures with respect to the coupled condition.

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…

Spin-harmonic structures on low-dimensional manifolds.

SU(n)-structures derived from Kähler manifolds with torus actions.

We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2.

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

Study characterizes SU(3)-structures with special Lagrangian 3-folds as minimal.

The set of maximal non-integrable structures $(SU(2)\times SU(2),B,I)$, where $B$ is Killing-Cartan metric is described as subset of $\mathbb{CP}^3$. The visualization of complex projective space $\mathbb{CP}^3$ as tetrahedron which edges and faces are $\mathbb{CP}^1$ and $\mathbb{CP}^2$ is used.

There exist non-degenerate 3-form $dω_I$, $ω_I(X,Y)=g(IX,Y)$, for each leftinvariant almost Hermitian structure $(g,I)$, where $g$ is Killing-Cartan metric on the $M=S^3\times S^3=SU(2)\times SU(2)$. Known \cite{H1}, that arbitrary non-degenerate 3-form on the 6-dimensional manifold, with some additional properties def…

No left-invariant hypercomplex structures found on compact Lie groups.

SU(3)-structures found on specific orbifold quotients.

We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…

We consider 5-manifolds with a contact form arising from a hypo structure, which we call \emph{hypo-contact}. We provide conditions which imply that there exists such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field $X$ preserv…

The Riemannian symmetric space SU_{2,m}/S(U_2U_m) is both Hermitian symmetric and quaternionic Kahler symmetric. Let M be a hypersurface in SU_{2,m}/S(U_2U_m) and denote by TM its tangent bundle. The complex structure of SU_{2,m}/S(U_2U_m) determines a maximal complex subbundle C of TM, and the quaternionic structure o…

We present a comprehensive classification of supersymmetric vacua of M-theory compactification on seven-dimensional manifolds with general four-form fluxes. We analyze the cases where the resulting four-dimensional vacua have N = 1,2,3,4 supersymmetry and the internal space allows for SU(2), SU(3) or G_2 structures. In…

Study of SU(2)-structures on tangent sphere bundles, discovering new metrics and structures.

Study cohomology spaces of left-invariant involutive structures on SU(2).

Study connects complex and symplectic supersymmetric systems on SU(4)-structure backgrounds.

SU(3)-structures found on quotients of 3-Sasakian manifolds.

One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gaug…

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 propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)…

We study the moduli space of SU(3) structure manifolds X that form the internal compact spaces in four-dimensional N=1/2 domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, X forms a non-compact 7-manifold Y with torsionful G2 structu…

In anlogy with the work of R. Bryant on the Ricci tensor of a G$_2$-structure, we study the intrinsic torsion of an SU$(2)$-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove…

The paper studies a flow on warped G2-structures, providing conditions for existence and describing new solutions.

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…

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 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…

Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.

New invariants show distinct contact structures have unique instanton homology.

It is shown that the compact Lie group SU(3) admits an Sp(2)Sp(1)-structure whose distinguished 2-forms $ω_1,ω_2,ω_3$ span a differential ideal. This is achieved by first reducing the structure further to a subgroup isomorphic to SO(3).

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent submodules we establish simple conditions under which an invariant metric f-structure (f,…

Paper proves non-existence of certain balanced metrics on six-manifolds.

This article can be viewed as a continuation of the articles arXiv:0912.3486 and arXiv:1012.3714 where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which ad…

Advances geometric structure flows, proving short-time existence and uniqueness for various flows.

Study classifies 8D Lie groups with specific foliations and Hermitian structures.