Sasakian manifolds, String models, Geometry, Riemannian
We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2n(2n+1), equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S3. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π1(M).
Ghosh, Amalendu and Sharma, Ramesh, "Sasakian Manifolds with Purely Transversal Bach Tensor" (2017). Mathematics Faculty Publications. 11.
Ghosh, Amalendu, and Ramesh Sharma. "Sasakian manifolds with purely transversal Bach tensor." Journal of Mathematical Physics 58, no. 10 (2017): 103502. doi:10.1063/1.4986492