A mainstream in mathematics is to study the relation between the geometry and topology of a manifold. The geometry is about the distance, length, area, and volume and is determined by the curvature. The topology is about properties that are preserved under continuous deformations. It is generally observed that a suitable assumption on the geometry would lead to certain restrictions on the topology. In this talk, we’ll discuss some recent progress in this direction, particularly from the perspective of the curvature of the second kind. There will be a resolution of a conjecture proposed by S. Nishikawa and rigidity results for gradient Ricci solitons. The talk is based on joint work with X. Cao, X. Cao-M. Gursky, and X. Cao- E. Ribeiro Jr.