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Some topics in Sasakian geometry, a survey

Aleksy Tralle

Abstract

In the seminal book of Boyer and Galicki "Sasakian Geometry" the authors formulated a research program of studying topological properties and answering questions about the existence of Sasakian structures. We survey recent progress in this topic.

Some topics in Sasakian geometry, a survey

Abstract

In the seminal book of Boyer and Galicki "Sasakian Geometry" the authors formulated a research program of studying topological properties and answering questions about the existence of Sasakian structures. We survey recent progress in this topic.
Paper Structure (25 sections, 32 theorems, 75 equations)

This paper contains 25 sections, 32 theorems, 75 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Sasakian and K-contact manifolds
  3. Formal manifolds and Massey products
  4. Tievsky's model
  5. Quasi-regular Sasakian manifolds as Seifert bundles over cyclic orbifolds
  6. Symplectic and Kähler singular orbifolds and Seifert bundles
  7. Constructions of cyclic Kähler orbifolds and Seifert bundles
  8. Fundamental groups of total spaces of Seifert bundles
  9. The sum up of the construction
  10. Existence problems for Smale-Barden manifolds
  11. Smale-Barden manifolds
  12. Definite Sasakian structures
  13. Rational homology spheres
  14. Quasi-regular negative Sasakian structures on $\#_k(S^2\times S^3)$
  15. Realization of homological data by Sasakian manifolds in dimension 5
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(M,\eta\, , \xi\, , \phi\, ,g)$ be a compact Sasakian manifold of dimension $2n+1$. Then, the Betti number $b_{r}(M)$ and the basic Betti number $b_{r}^{B}(M)$ are related by for $0\,\leq\, r\,\leq\, n$, In particular, if $r$ is odd and $r\,\leq\, n$, then $b_{r}(M)\,=\,b_{r}^{B}(M)$.

Theorems & Definitions (44)

  • Theorem 1.1: BG
  • Theorem 1.2
  • Proposition 1.3: R
  • Theorem 1.4: DGMSTO
  • Theorem 1.5: Tievsky
  • Definition 2.1: M
  • Definition 2.2
  • Theorem 2.3: M
  • Definition 2.4
  • Theorem 2.5: BG
  • ...and 34 more