First you have to know that many companies do not have a business analyst, and when a business analyst exists, then most (if not all) of his responsibilities usually are also the responsibilities of the project manager. The business analyst exists to take some load off the shoulders of the project manager (to whom he reports to).
Now here's a list of all the shared (or common) activities between the business analyst and the project manager:
- Gathering the requirements from the customer/client
- Managing the client's expectations
- Managing change requests
- Managing the traceability matrix
It is important to note that the business analyst (when the role exists) does exactly what the project manager does, but at a more granular (detailed) level.