Geometric Formality

我们将研究紧致定向流形上的黎曼度量，其中所有调和形式的楔形积都是调和的。在小维度上，我们的目标是对这些指标进行分类。在任意维中，这种度量的调和形式具有非常特殊的性质，并定义了有趣的几何结构，如叶理和辛结构。我们调查均匀的例子，和那些接近均匀的（例如，cohomogeneity一个，或bifurcents）。进一步，我们通过辛几何研究这种度量的结构。在辛几何中，我们研究了Brylinski意义下的辛调和形式而不是Hodge理论意义下的调和形式的相似性质。

We shall investigate Riemannian metrics on compact oriented manifolds for which all wedge products of harmonic forms are harmonic. In small dimensions, we aim to classify these metrics. In arbitrary dimensions, forms harmonic for such a metric have very special properties and define interesting geometric structures, like foliations and symplectic structures. We investigate homogeneous examples, and those close to homogeneous (e.g. of cohomogeneity one, or biquotients). Further we study constructions of such metrics via symplectic geometry. Within symplectic geometry, we study the analogous property for symplectically harmonic forms in the sense of Brylinski instead of the harmonic forms in the sense of Hodge theory.