In this paper, we would like to make a report on Morse inequalities {and embeddings for CR manifolds with transversal CR circle action}. The results are contained in [24], [23], [21], [20] and [25]. Furthermore, in the last section we will give an explicit example for $S^1$-equivariant Szego kernels and their expansion on the sphere $S^3$ in $\mathbb C^2$ with respect to a family of transversal CR $S^1$-actions.

Received: 2020-03-19
Revised :
Accepted: 2020-06-15