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Discovery Projects - Grant ID: DP130103485

Title
Discovery Projects - Grant ID: DP130103485
Funding
ARC | Discovery Projects
Contract (GA) number
DP130103485
Start Date
2013/01/01
End Date
2015/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP130103485

 

  • Singular multicontact structures

    Ottazzi, Alessandro; Schmalz, Gerd (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.

    On the generalization of Forelli's theorem

    Joo, Jae-Cheon; Kim, Kang-Tae; Schmalz, Gerd (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forelli's theorem on the complex analyticity of the functions that are: (1) $\mathcal{C}^\infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.

    Curvature properties of metric nilpotent Lie algebras which are independent of metric

    Cairns, G.; Galić, A. Hinić; Nikolayevsky, Y. (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on $\mathfrak{g}$. In the second part we study the subsets of $\mathfrak{g}$ which are, for some inner product, the eigenvectors of the Ricci operator...

    Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties

    Matveev, Vladimir S.; Nikolayevsky, Yuri (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is closed. We further prove a de Rham type splitting theorem which states that if such a manifold is analytic, it is isometric to the Riemannian product of a Euclidean space and an in...

    Solvable Lie groups of negative Ricci curvature

    Nikolayevsky, Y.; Nikonorov, Yu. G. (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, and discuss some open questions.

    Explicit description of spherical rigid hypersurfaces in C^2

    Ezhov, Vladimir; Schmalz, Gerd (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature condition for rigid hypersurfaces.

    Solvable extensions of negative Ricci curvature of filiform Lie groups

    We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, when $\mathfrak{n}$ is one of the algebras of rank two, $L_n$ or $Q_n$, and $\mathfrak{g}$ is a one-dimensional extension of $\mathfrak{n}$, in which cases the conditions are give...

    On invariant Riemannian metrics on Ledger-Obata spaces

    Nikolayevsky, Y.; Nikonorov, Yu. G. (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We study invariant metrics on Ledger-Obata spaces $F^m/\mathrm{diag}(F)$. We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case $m=3$, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an eff...

    Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over $\mathbb{Z}_2$

    Nikolayevsky, Yuri; Tsartsaflis, Ioannis (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$ are isomorphic, in contrast with the case of the ground field of characteristic zero, and we obtain a complete description of them. In the finite-dimensional case, we find the first three Betti numbers of $\mathfrak{m}_0(n)$ an...

    $H$-contact unit tangent sphere bundles of Riemannian manifolds

    Nikolayevsky, Yuri; Park, Jeong Hyeong (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP130103485 (DP130103485)
    A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.
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