arXiv research

Study shows some biquotients have non-biquotient tangent bundles.

New findings on biquotient bundles lacking inverses in various dimensions.

We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain s…

New method calculates signatures of biquotients.

A closed manifold is called a biquotient if it is diffeomorphic to K\G/H for some compact Lie group G with closed subgroups K and H such that K acts freely on G/H. Biquotients are a major source of examples of Riemannian manifolds with nonnegative sectional curvature. We prove several classification results for biquoti…

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

We classify all compact simply connected biquotients of dimension 4 and 5. In particular, all pairs of groups $(G,H)$ and embeddings $H\rightarrow G\times G$ giving rise to a particular biquotient are classified.

We classify all compact simply connected biquotients of dimension 6 and 7. For each $6$-dimensional biquotient, all pairs of groups $(G,H)$ and homomorphisms $H\rightarrow G\times G$ giving rise to it are classified.

We classify all compact simply connected biquotients of the form $G/\!\!/ SU(2)^2$ for $G =SU(4), SO(7), Spin(7)$, or $G = \mathbf{G}_2\times SU(2)$. In particular, we show there are precisely $2$ inhomogeneous reduced biquotients in the first and last case, and $10$ in the middle cases.

We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.

One of the main methods of constructing new spaces with positive or almost positive curvature is the study of biquotients first studied in detail by Eschenburg. We classify orbifold biquotients of the Lie Group $SU(3)$, and construct a new example of a 5-dimensional orbifold with almost positive curvature. Furthermore,…

We construct symplectic structures on roughly half of all equal rank biquotients of the form $G//T$, where $G$ is a compact simple Lie group and $T$ a torus, and investigate Hamiltonian Lie group actions on them. For the Eschenburg flag, this action has similar properties as Tolman's and Woodward's examples of Hamilton…

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another are the circle bundles over a base, which itself is one of the family of CP^1 b…

We show there are precisely 15 inhomogeneous biquotients of the form $Sp(3)// Sp(1)^2$ and show that at least 8 of them admit metrics of quasi-positive curvature.

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rat…

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschen…

These are notes of a talk I gave in a seminar at the University of Pennsylvania summarizing results in the Habilitation by Jost Eschenburg on "Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekruemmten Orbitraeumen". Due to the fact that it is published in a not easily accesible journal (and is writt…

Suppose $φ_3:Sp(1)\rightarrow Sp(2)$ denotes the unique irreducible $4$-dimensional representation of $Sp(1) = SU(2)$ and consider the two subgroups $H_1, H_2\subseteq Sp(3)$ with $H_1 = \{\operatorname{diag}(φ_3(q_1), q_1): q_1 \in Sp(1)\}$ and $H_2 = \{\operatorname{diag}(φ_3(q_2),1):q_2\in Sp(1)\}$. We show that the…

We characterise simply-connected biquotients which potentially admit metrics of holonomy G_2. We prove that there are at most three real homotopy types of rationally elliptic such manifolds---all of them being formal. In the course of this examination we classify rationally elliptic homotopy types and characterise 7-di…

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus $T^{n-2}$ is equivariantly diffeomorphic to an isometric action on a normal biqu…

The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the iso…

An $n$-dimensional manifold $M$ is said to be rationally $4$-periodic if there is an element $e\in H^4(M;\mathbb{Q})$ with the property that cupping with $e$, $\cdot \cup e:H^\ast(M;\mathbb{Q})\rightarrow H^{\ast + 4}(M;\mathbb{Q})$ is injective for $0< \ast \leq \dim M-4$ and surjective when $0\leq \ast < \dim M-4$. W…

We prove that any base space of Riemannian submersion from a compact Lie group (with bi-invariant metric) must have a basic property previously known for normal biquotients; namely, any zero-curvature plane exponentiates to a flat.

Gromoll and Meyer have represented a certain exotic 7-sphere $Σ^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Σ^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ wh…

We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by $SU(2)$ or $SO(3)$. We show that their Euler characteristic agrees with that of the known examples, i.e. $S^6$, $\mathbb{CP}^3$, the Wallach space $SU(3)/T^2$ and the bi…

We discuss the question of geometric formality for rationally elliptic manifolds of dimension $6$ and $7$. We prove that a geometrically formal six-dimensional biquotient with $b_{2}=3$ has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with $b_2\leq 2$ and …

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…

The authors examine topological properties of the 7-dimensional Eschenburg biquotients diag(z^k1,z^k2,z^k3)\SU(3)/diag(z^l1,z^l2,z^l3). A subfamily of these spaces carry a 3-Sasakian metric. The authors show that among this subfamily there exist many 3-Sasakian spaces which are homeomorphic but not diffeomorphic. In ad…

The study of quotient structures in multi-graded bundles, including double vector bundles.

Study of semi-principal bundles using group actions and wreath products.

Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

Introduces Darboux-Lie derivative for fiber bundles.

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…

We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples. It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and conne…

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…

Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.

Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.

Identifies images of determinant morphism for specific co-Higgs bundles.

Bundling of graph edges (node-to-node connections) is a common technique to enhance visibility of overall trends in the edge structure of a large graph layout, and a large variety of bundling algorithms have been proposed. However, with strong bundling, it becomes hard to identify origins and destinations of individual…

The paper defines and explores vector bundles that can be turned and their properties.

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.

Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.

Generalizes Higgs bundles theory using a vector bundle twist.