arXiv research

Geodesic orbit metrics proven on specific homogeneous spaces.

Study the smallest Laplace eigenvalue in special geometric spaces.

This paper is devoted to a systematic study and classification of invariant affine or metric connections on certain classes of naturally reductive spaces. For any non-symmetric, effective, strongly isotropy irreducible homogeneous Riemannian manifold $(M=G/K, g)$, we compute the dimensions of the spaces of $G$-invarian…

The study classifies homogeneous manifolds with specific geometric properties.

We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while …

We classify homogeneous pseudo-Riemannian manifolds of index 4 which admit an invariant almost hyper-Hermitian structure and an H-irreducible isotropy group. The main result is that all these spaces are flat except in dimension 12.

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…

We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible …

Study extends reflective submanifold theory to compact homogeneous spaces.

Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.

Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(ν_i, Φ_i)$ be an irreducible factor of the normal holonomy representation $(ν_pM, Φ(p))$. We prove that there exists a basis of a section $Σ_i\subset ν_i$ of $Φ_i$ such that the corresponding shape operators have rational…

We consider the question: can the isotropy representation of an irreducible pseudo-Riemannian symmetric space be realized as a conformal holonomy group? Using recent results of Cap, Gover and Hammerl, we study the representations of SO(2,1), PSU(2,1) and PSp(2,1) as isotropy groups of irreducible symmetric spaces of si…

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces $G/H$ with two irreducible submodules in…

The study classifies strongly irreducible Heegaard splittings in hyperbolic 3-manifolds.

For a compact homogeneous space $G/K$, we study the problem of existence of $G$-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of $G$. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, w…

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.

We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of sp…

Study solves Ricci curvature problem for specific noncompact spaces.

The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading …

Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.

We show that a strongly irreducible and boundary-strongly irreducible surface can be isotoped to be almost normal in a triangulated 3-manifold.

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility a…

Study on Einstein metrics on specific homogeneous spaces.

We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant Einstein…

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

Extends particle classification to curved space-times using groupoids.

We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy representation of the principal orbit consists of exactly three distinct i…

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard split…

It has been shown elsewhere that a strongly irreducible Heegaard splitting surface Q of a compact orientable 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M.

We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained from isotropy representations of certain quaternion-Kähler symmetric spaces by restricting to …

We consider the eigenfunctions of the Laplace operator $Δ$ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $Δ$ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M…

We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings of a graph manifold o…

We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic wi…

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence…

Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces S_i of these strongly irreducible Heegaard splittings have t…

A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general proble…

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable oriented double cover of a non-orientable minimal surface with a vertical handle atta…

We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag m…

We study the way a strongly irreducible Heegaard surface $Σ$ intersects a knot exterior $X$ embedded in a 3-manifold, and show that if $Σ\cap \partial X$ consists of simple closed curves which are essential in both $Σ$ and $\partial X$, then the intersection $X \cap Σ$ consists of meridional annuli only. As an applicat…

We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1. We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irr…

This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. T…

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or…

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric space…

We show that if an orientable Seifert fibered space $M$ with an orientable genus $g$ base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of $\mathbf{Z}^{2g}$. The corr…

Lower bound on volumes of special mapping tori.

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $Ad_K$-invariant irreducible summands, the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These e…