Homogeneous spaces with invariant Koszul-Vinberg structures. (English) Zbl 1527.53049
This article is primarily devoted to providing an algebraic characterization of \(G\)-invariant Koszul-Vinberg structures on \(G\)-homogeneous manifolds. It relies to the spirit of Nomizu’s theorem on invariant connections, as referenced in [K. Nomizu, Am. J. Math. 76, 33–65 (1954; Zbl 0059.15805); A. Elduque, Commun. Math. 28, No. 2, 199–229 (2020; Zbl 07300190)]. The authors give a list of examples, thus providing invariant Koszul-Vinberg structures on both reductive pairs and symmetric spaces. Besides, the article presents several examples of circumstances where the Koszul-Vinberg structures are pseudo-Hessian. A significant result suggests that if the Lie group \(G\) is semi-simple, \(G/H\) will not carry any non-trivial \(G\)-invariant pseudo-Hessian structure. This is explicitly formulated in Theorem 5.7. As a direct consequence of this result, the authors obtain a new proof of a result by H. Shima [The geometry of Hessian structures. Hackensack, NJ: World Scientific (2007; Zbl 1244.53004), Theorem 9.2]. Another remarkable result of the paper reveals that the leaves of the affine foliation, which are linked to an invariant Koszul-Vinberg structure, are homogeneous pseudo-Hessian manifolds.
Reviewer: Zhuo Chen (Beijing)
MSC:
| 53C30 | Differential geometry of homogeneous manifolds |
| 53B05 | Linear and affine connections |
| 53C12 | Foliations (differential geometric aspects) |
| 17D25 | Lie-admissible algebras |
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