*Charles P. Boyer and Krzysztof Galicki*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter begins the study of Sasakian geometry, where all important structure theorems are presented. It then gathers all the results concerning the geometry, topology, and curvature properties ...
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This chapter begins the study of Sasakian geometry, where all important structure theorems are presented. It then gathers all the results concerning the geometry, topology, and curvature properties of both K-contact and Sasakian manifolds. The relation between Sasakian and algebraic geometry is stressed, as well as important invariants such as the basic cohomology groups, and the basic first Chern class that are associated to certain spaces of Sasakian structures. A main tool used in the text is the transverse Yau Theorem due to El Kacimi-Alaoui.Less

This chapter begins the study of Sasakian geometry, where all important structure theorems are presented. It then gathers all the results concerning the geometry, topology, and curvature properties of both K-contact and Sasakian manifolds. The relation between Sasakian and algebraic geometry is stressed, as well as important invariants such as the basic cohomology groups, and the basic first Chern class that are associated to certain spaces of Sasakian structures. A main tool used in the text is the transverse Yau Theorem due to El Kacimi-Alaoui.

*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ...
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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.Less

This chapter deals with topological invariants and differential geometry. It first considers a topological space *X* for which singular homology and cohomology are defined, along with the Euler number *e*(*X*). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space *X*. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold *X*.