论文标题
紧凑型同质空间上的谐波3形
Harmonic 3-forms on compact homogeneous spaces
论文作者
论文摘要
研究了紧凑型均匀空间的第三个实际DE RHAM共同体。给定的$ m = g/k $，带有$ g $ compact的半imple，我们首先表明，$ \ mathfrak {g} $上的每个双重对称的对称的双线性双线性$ q $ $ m $，扮演所谓的cartan $ 3 $ -form $ q（[\ cdot，\ cdot]，\ cdot）$上的$ 3 $ - form $ q。实际上，$ h^3（g/k）$中的每个类都有一个独特的代表性$ h_q $。其次，专注于具有最丰富的第三个同提的同质空间（除了谎言组），即$ b_3（g/k）= s-1 $如果$ g $具有$ s $ s $简单的因素，我们给出了$ q $和$ q $ g $ g $ g $ $ g $的条件，以$ h_q $ $ h_q $ g $ g g $ g g $ g g $ h_q $ - $ g/k $。作为一个应用程序，我们得到任何$ 3 $ -form $ h_q $与标准度量的谐音，尽管对于任何其他普通度量，但只有一个$ h_q $ to缩放为谐波。此外，在合适的$（2S-1）$ - $ g $ - invariant的参数家族中，我们证明，如果$ \ mathfrak {k} $是abelian的情况，同一行为会发生相同的行为：每个$ h_q $ as co $ g $ harmonic co $ harmonic cop- $ g $ $ g $ $ $ $ $ $ - $ $ - $ - 33 33 3 3 3 3 3 3 3 3 33缩放）。在$ \ m athfrak {k} $不是Abelian时，每个$ h_q $的特殊指标都是$ g $ - harmonic取决于$ 3 $参数。
The third real de Rham cohomology of compact homogeneous spaces is studied. Given $M=G/K$ with $G$ compact semisimple, we first show that each bi-invariant symmetric bilinear form $Q$ on $\mathfrak{g}$ such that $Q|_{\mathfrak{k}\times\mathfrak{k}}=0$ naturally defines a $G$-invariant closed $3$-form $H_Q$ on $M$, which plays the role of the so called Cartan $3$-form $Q([\cdot,\cdot],\cdot)$ on the compact Lie group $G$. Indeed, every class in $H^3(G/K)$ has a unique representative $H_Q$. Secondly, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., $b_3(G/K)=s-1$ if $G$ has $s$ simple factors, we give the conditions to be fulfilled by $Q$ and a given $G$-invariant metric $g$ in order for $H_Q$ to be $g$-harmonic, in terms of algebraic invariants of $G/K$. As an application, we obtain that any $3$-form $H_Q$ is harmonic with respect to the standard metric, although for any other normal metric, there is only one $H_Q$ up to scaling which is harmonic. Furthermore, among a suitable $(2s-1)$-parameter family of $G$-invariant metrics, we prove that the same behavior occurs if $\mathfrak{k}$ is abelian: either every $H_Q$ is $g$-harmonic (this family of metrics depends on $s$ parameters) or there is a unique $g$-harmonic $3$-form $H_Q$ (up to scaling). In the case when $\mathfrak{k}$ is not abelian, the special metrics for which every $H_Q$ is $g$-harmonic depend on $3$ parameters.
