A note on the volume entropy of harmonic manifolds of hypergeometric type
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909796307828736 |
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| author | Satoh, Hiroyasu |
| author_facet | Satoh, Hiroyasu |
| contents | Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature $\mathrm{Ric} = -(n-1)$, we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature $-1$, while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05896 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the volume entropy of harmonic manifolds of hypergeometric type Satoh, Hiroyasu Differential Geometry 53C21, 53C25, 53C20 Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature $\mathrm{Ric} = -(n-1)$, we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature $-1$, while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases. |
| title | A note on the volume entropy of harmonic manifolds of hypergeometric type |
| topic | Differential Geometry 53C21, 53C25, 53C20 |
| url | https://arxiv.org/abs/2405.05896 |