# A Note On Smale Manifolds And Lorentzian Sasaki-Einstein Geometry

## Document Type

Article

## Publication Date

2016

## Published In

Bulletin Mathematique De La Societe Des Sciences Mathematiques De Roumanie

## Abstract

In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds M. It has already been established by the author that such metrics exist on the so-called torsion free Smale manifolds, i.e. the k-fold connected sum of S² × S³ for all k. Now, we show that LSE metrics exist on Smale manifolds for which H₂(M, Z)tor is nontrivial. In particular, we show that most simply-connected positive Sasakian rational homology 5-spheres are also negative Sasakian (hence LSE). Moreover, we show that for each pair of positive integers (n, s) with n, s > 1, there exists a Lorentzian Sasaki-Einstein Smale manifold M such that H2(M, Z)tors = (Z/n)2s . Finally, we are able to construct so-called mixed Smale manifolds (connect sum of torsion free Smale manifolds with rational homology spheres) which admit LSE metrics and have arbitrary second Betti number. This gives infinitely many examples which do not admit positive Sasakian structures. These results partially address open problems formulated by C. Boyer and K. Galicki.

## Keywords

Lorentzian-Einstein metrics, rational homology spheres Sasakian manifolds

## Recommended Citation

Ralph R. Gomez.
(2016).
"A Note On Smale Manifolds And Lorentzian Sasaki-Einstein Geometry".
*Bulletin Mathematique De La Societe Des Sciences Mathematiques De Roumanie.*
Volume 59,
Issue 2.
175-185.

https://works.swarthmore.edu/fac-math-stat/182