Generalised Raychaudhuri equations for strings in the presence of an antisymmetric tensor field

The generalised Raychaudhuri equations derived by Capovilla and Guven are exclusively for extremal, timelike Nambu--Goto membranes. In this article, we construct the corresponding equations for string world--sheets in the presence of a background Kalb--Ramond field. We analyse the full set of equations by concentrating on special cases in which the generalised shear or the generalised rotation or both are set to zero. If only the generalised shear is set to zero then it is possible to identify the components of the generalised rotation with the projections of the field strength of the Kalb--Ramond potential.