The aim of this paper is to address some results closely related to the conjecture of Kosniowski about the number of fixed points on a unitary $S^1$-manifold with only isolated fixed points. More precisely, if certain $S^1$-equivariant Chern characteristic number of a unitary $S^1$-manifold $M$ is non-zero, we give a sharp (in certan cases) lower bound on the number of isolated fixed points in terms of certain integer powers in the $S^1$-equivariant Chern number. In addition, we also deal with the case of oriented unitary $T^n$-manifolds.
"On the conjecture of Kosinowski." Asian J. Math. 16 (2) 271 - 278, June 2012.