D. Kotschick, S. Terzic
We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.
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Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that $(\mathbb{Z}_2\oplus \mathbb{Z}_2)$-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.
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 sphere bundles.
We prove several long-standing conjectures by Chen--Nagano on cohomological descriptions of the cardinalities of maximal antipodal sets in symmetric spaces. We actually extend these conjectures to the setting of generalised symmetric spaces of finite abelian $p$-groups and verify them mostly in this broader context drawing upon techniques from equivariant cohomology theory.
We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic Kähker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose total space is non-formal.
We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-genera…
We use generalised cross--ratios to prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of the boundary of symmetric Riemannian spaces of rank 1 and of negative curvature.
We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters $k=2$ and $k=3$ naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric $(2,3,5)$-distribution (described locally by a certain function $φ(x,q)=\frac{q^2}{H''(x)}$) to a Ricci-flat one.
We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A $C^0$ \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metrics to recover the existence of solutions in some of the known cases. Known results are then applied to an analytic problem arising from Chern-Weil theory and to a special Lagrangian-type equation arising from mirror symmetry.
Components of generalised complex structures on transitive Courant algebroids
研究transitive Courant algebroids上的广义复结构，通过其组成部分分析其可积性，并与泊松结构和辛结构相关联。
We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form $D= \nabla + Ω$ in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. Specialising this to the case of symmetric M-theory backgrounds (i.e. $(M,g,F)$ with $(M,g)$ a symmetric space and $F$ an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric s…
Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps
本文研究带广义 $φ$-尖点的完备黎曼流形，在紧致群作用下建立 Dirac 算子等变指标定理，并刻画尖点贡献与谱对称性及 $η$-不变量的关系。
A Calabi operator for Riemannian locally symmetric spaces
本文将常曲率流形上的卡拉比算子推广到黎曼局部对称空间，证明其在不可约情形下总能为Killing算子值域提供二阶局部可积性条件，并刻画了乘积情形下算子失效的精确条件。
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of ``nonelementary'' discrete isometry groups of higher rank symmetric spaces. We obtain…
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