Viviana del Barco, Andrei Moroianu
We study left-invariant Killing forms of arbitrary degree on simply connected $2-$step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.
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We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.
We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases,…
Magnetic Killing tensors and first integrals of the magnetic flow
本文提出磁 Killing 对称张量，用于构造磁流的首个积分，并证明某些 nilmanifolds 上磁流的可积性。
We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.
Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate. After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent group…
We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of $d ω$ and its Hodge dual $* dω$ where $ω$ is the fundamental 2-form of the nearly Kähler structure. The proof is based on a fundamental integrability condition for conformal Killing forms. We have partial results in the case of conformal Killing 2-forms. In particular we show the non-existence of J-anti-invariant Killing 2-forms.
The Koutras-McIntosh family of metrics include conformally flat pp-waves and the Wils metric. It appeared in a paper of 1996 by Koutras-McIntosh as an example of a pure radiation spacetime without scalar curvature invariants or infinitesimal symmetries. Here we demonstrate that these metrics have no "hidden symmetries", by which we mean Killing tensors of low degrees. For the particular case of Wils metrics we show the nonexistence of Killing tensors up to degree 6. The technique we use is the geometric theory of overdetermined PDEs and the Cartan prolongation-projection method. Application of…
Conformal Killing forms in Kaehler geometry
本文在Kähler几何中显式求解中间度数的共形Killing形式方程，完成紧致情形的分类，并首次给出非Hamiltonian 2-形式来源的实例，基于Calabi型流形，同时提供Hermitian Killing形式的结构与例子。
On the eta invariant of (2,3,5) distributions
本文研究闭五流形上一般秩二分布相关的Rumin复形中Heisenberg阶二自伴算子的eta不变量，证明(2,3,5)幂零流形上eta不变量为零但eta函数非平凡，并给出其用基本函数表达的公式。
Nilpotent Lie algebras obtained by quivers and Ricci solitons
本研究提出一种利用无环有限箭图的路径概念构造幂零李代数的方法，证明对应单连通幂零李群容许左不变里奇孤立子，可构造任意高幂零度的里奇孤立子幂零流形族。
Bogomolov-Guan Manifolds are not formal
本文证明所有复维数至少为四的 Bogomolov--Guan 流形在有理数域上非形式，通过构造一致的八次 a-Massey 积并利用 Saito 分解定理，还确定了其第三 Betti 数为 3。
We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a $k$-degree form $0\leq k\leq 4$, those modified with multi-degree $(0,k)$-form $0\leq k\leq 3$ and the horizon Dirac operators which are modified with a multi-degree $(1,2,4)$-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero…
We construct a natural conformally invariant one-form of weight $-2k$ on any $2k$-dimensional pseudo-Riemannian manifold which is closely related to the Pfaffian of the Weyl tensor. On oriented manifolds, we also construct natural conformally invariant one-forms of weight $-4k$ on any $4k$-dimensional pseudo-Riemannian manifold which are closely related to top degree Pontrjagin forms. The weight of these forms implies that they define functionals on the space of conformal Killing fields. On Riemannian manifolds, we show that this functional is trivial for the former form but not for the latter…
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