Narrow Results

Smale–Barden manifolds $M$ are classified by their second homology $H_2(M,\mathbb{Z})$ and the Barden invariant $i(M)$. It is an important and difficult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all $M$ with $H_2(M,\mathbb{Z})={\mathbb{Z}}^k\oplus ({\oplus }_{i=1}^r{\mathbb{Z}}_{m_i}^{2g_i})$ and $i(M)=0,\infty ,$ provided that $k\ge 1$, $m_i\ge 2,g_i\ge 1$, $m_i$ are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.

We show that the scalar curvature of a $K$-contact Ricci soliton is constant and satisfies sharp bounds. Next we show that the scalar curvature of a ($2n+1$)-dimensional $K$-contact Ricci almost soliton is equal to $2n(2n+1)$ plus the divergence of a global vector field. Finally, we show that, if a complete connected Sasakian or $\eta$-Einstein $K$-contact manifold of dimension $>3$ is a proper Ricci almost soliton, then it is isometric to a unit sphere.