All of the calls into normal java objects[0] from the stateless bean
occur in the same transaction context.  However, normal java objects
aren't inherently transaction-aware -- if you need to manipulate
transactional resources from a normal java object, it's up to you to
provide those resources to the object in a form it understands.

Particularly, RuntimeException and
@ApplicationException(rollback=true) exceptions only cause transaction
rollback when they cross a transaction boundary (like an EJB call), so
if the Data Access object in your diagram throws a RuntimeException
that gets caught in the Business Logic object, no transaction rollback
will occur.

Your DBManager will take part in the same transaction if, and only if,
it either gets passed a Connection or DataSource obtained within the
transaction (either injected into the session bean or obtained from
JNDI) or it obtains one itself via JNDI (either directly or
indirectly).

If any of those objects calls a session bean in the same container (or
if your container supports transaction propagation between servers)
then the called bean will participate in the same transaction
(assuming it's marked as either REQUIRES or SUPPORTS).  Otherwise, if
it's REQUIRES (and the transaction can't be propagated) or
REQUIRES_NEW, a new transaction will begin for that call only.

[0] I detest the moniker "POJO".