arXiv research

Study on conformal transformations of Cahen-Wallach spaces, focusing on fixed points and discontinuous groups.

We classify generalized Wallach spaces which are g.o. spaces. We also investigate homogeneous geodesics in generalized Wallach spaces for any given invariant Riemannian metric and we give some examples.

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these paramete…

Study of a 3D system on Wallach spaces, finding interrelations with invariant metrics.

In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.

Curved metrics on Wallach spaces bounded by curves under flow.

The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.

Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.

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

Invariant Einstein metrics on generalized Wallach spaces have been classified except $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$. In this paper, we give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type $SO(k+l+m)/SO(k)\times SO(…

An infinite family of distinct harmonic morphisms with minimal circle fibers from the 7-dimensional homogeneous Allof-Wallach spaces of positive curvature onto the 6-dimensional flag manifolds is given.

Normalized Ricci flow preserves positivity of Ricci curvature on some generalized Wallach spaces.

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers $a_1, a_2, a_3$. Our interest is to determine the type of sing…

This paper classifies 13D manifolds based on Bazaikin spaces.

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…

In this paper, we obtain two-sided bounds for the volumes of the Aloff-Wallach spaces $W(p,q),$ compute maximal and minimal sectional curvature for the spaces $W(n,n+1),$ and use this information to estimate the injectivity radii: We derive an upper bound for the injectivity radii of $W(p,q)$ and a lower bound for the …

Generalizes embedding complex Grassmannians into quadrics.

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

Study second-order obstruction to nearly $G_2$ structure deformations.

Geodesic completeness proven for certain Lorentzian spaces.

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

We systematically discuss connections on the spinor bundle of Cahen-Wallach symmetric spaces. A large class of these connections is closely connected to a quadratic relation on Clifford algebras. This relation in turn is associated to the symmetric linear map that defines the underlying space. We present various soluti…

The paper finds new $\mathrm{Spin}(7)$ metrics with specific orbits.

We study topological structures of the sets $(0,1/2)^3 \cap Ω$ and $(0,1/2)^3 \setminus Ω$, where~$Ω$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These problems arise in studying of general properties of degenerate singular points of dynamical systems ob…

Ricci flow can change metrics with positive curvature to those without.

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 Ricci flow on specific homogeneous spaces.

Characterizes equigeodesics on specific homogeneous spaces.

The study characterizes and verifies equivariant embeddings of symmetric Kählerian manifolds.

Causal properties of Lorentzian symmetric spaces are investigated in the paper. The global hyperbolicity of the Cahen--Wallach Lorentzian symmetric spaces is proved.

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type…

The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the …

We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metric…

New solutions found for G2 system on squashed manifolds.

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the …

We present an approach to solvable pseudo-Riemannian symmetric spaces based on papers of M.Cahen, M.Parker and N.Wallach. Thereby we reproduce the classification of solvable symmetric triples of Lorentzian signature $(1,n-1)$ and complete the case of signature $(2,n-2)$. Moreover we discuss the topology of non-simply-c…

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\fr…

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…

We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric wit…

New transitions found in Spin(7) holonomy metrics related to a dynamical system.

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in $\mathbb C^3$, $\mathbb H^3$ and $\mathbb{Ca}^3$. Together with our earlier work, this concl…

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one s…

The paper explores symplectic connections on homogeneous spaces, finding a unique invariant connection.