Harmonic 3-Forms on Compact Homogeneous Spaces
نویسندگان
چکیده
The third real de Rham cohomology of compact homogeneous spaces is studied. Given $$M=G/K$$ with G semisimple, we first show that each bi-invariant symmetric bilinear form Q on $${\mathfrak {g}} $$ such $$Q|_{{\mathfrak {k}} \times {\mathfrak }=0$$ naturally defines a G-invariant closed 3-form $$H_Q$$ M, which plays the role so called Cartan $$Q([\cdot ,\cdot ],\cdot )$$ Lie group G. Indeed, every class in $$H^3(G/K)$$ has unique representative . Second, focusing richest (other than groups), i.e., $$b_3(G/K)=s-1$$ if s simple factors, give conditions to be fulfilled by and given metric g order for g-harmonic, terms algebraic invariants G/K. As an application, obtain any harmonic respect standard metric, although other normal there only one up scaling harmonic. Furthermore, among suitable $$(2s-1)$$ -parameter family metrics, prove same behavior occurs abelian: either g-harmonic (this metrics depends parameters) or (up scaling). In case when not abelian, special depend 3 parameters.
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ژورنال
عنوان ژورنال: Journal of Geometric Analysis
سال: 2023
ISSN: ['1559-002X', '1050-6926']
DOI: https://doi.org/10.1007/s12220-023-01221-0