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Homogeneity and Inhomogeneity of Sasakian Geometries
We study homogeneous and inhomogeneous manifolds with various Sasakian geometries. First we provide a new and more illustrative proof of the classification of homogeneous 3-Sasaki manifolds, which was originally obtained by Boyer, Galicki and Mann [BGM]. In doing so we construct an explicit one-to-o...
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| Format: | Doctoral Thesis |
| Language: | English |
| Published: |
Philipps-Universität Marburg
2023
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| Online Access: | PDF Full Text |
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| Summary: | We study homogeneous and inhomogeneous manifolds with various Sasakian geometries. First we provide a new and more illustrative proof of the classification of homogeneous 3-Sasaki manifolds, which was originally obtained by Boyer, Galicki and Mann [BGM]. In doing so we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasaki manifolds and simple complex Lie algebras via the theory of root systems. These results also yield an alternative derivation of the classification of homogeneous positive quaternionic Kähler manifolds due to Alekseevskii [Alek]. Subsequently we apply similar techniques to degenerate 3-(α, δ)-Sasaki manifolds to deduce several results which limit the number of homogeneous spaces with this geometry. We prove that this category contains no non-trivial compact examples as well as exactly one family of nilpotent Lie groups, namely the quaternionic Heisenberg groups. By way of contrast we present a method to construct degenerate 3-(α, δ)-Sasaki manifolds as certain T^3-bundles over hyperkähler manifolds with integral Kähler classes, which is similar to the famous Boothby-Wang bundle [BW].The manifolds obtained this way are necessarily inhomogeneous and we develop a way to quantify "how far away from homogeneous" they are. To this end we utilize that Sasakian geometries naturally come with the so-called characteristic foliation and elaborate a generalization of the famous Bochner technique for foliations [Boch]. Later it turned out that this Bochner technique for foliations also follows from results in the article [HR], but the applications to Sasakian geometries are new. |
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| Physical Description: | 62 Pages |
| DOI: | 10.17192/z2023.0340 |