  We study the mutual interaction of a Bose-Einstein condensed gas with a
single mode of a high-finesse optical cavity. We show how the cavity
transmission reflects condensate properties and calculate the self-consistent
intra-cavity light field and condensate evolution. Solving the coupled
condensate-cavity equations we find that while falling through the cavity, the
condensate is adiabatically transfered into the ground state of the periodic
optical potential. This allows time dependent non-destructive measurements on
Bose-Einstein condensates with intriguing prospects for subsequent controlled
manipulation.
