In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds
It has already been established by the author that such metrics exist on the so-called torsion
free Smale manifolds, i.e. the -fold connected sum of
for all Now, we show that
LSE metrics exist on Smale manifolds for which
is nontrivial.
In particular, we show that most simply-connected positive Sasakian rational homology -spheres are
also negative Sasakian (hence LSE).
Moreover, we show that for each pair of positive integers with , there exists a Lorentzian Sasaki-Einstein Smale manifold
such that
. 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.

Key Words: Sasakian manifolds, Lorentzian-Einstien metrics, rational homology spheres.

2000 Mathematics Subject Classification: Primary: 53C25,

Secondary: 53D10, 53C50.