The steady thermocapillary convection in an axisymmetric liquid bridge in the absence of gravitational forces is considered. The relative change in the surface tension is assumed to be very small, i.e., the appropriate capillary number is much less than 1. A detailed analysis of the flow field is carried out for cylindrical bridges. The field equations are simplified by neglecting convective effects (Marangoni numberM=0, Reynolds number Re=0). The effects of the bridge aspect ratio, differential heating of the side walls, and arbitrary heating of the free surface are analyzed numerically. The results show that flow patterns consisting of one, two, three, or more columns of vortices, or of their superposition, may exist in the liquid, depending on the type of heating. Decrease of the aspect ratio (long bridges) results in an emergence of the core zones of almost parallel flow. Increase of the aspect ratio (short bridges) leads to the emergence of several layers of vortices, with the strength of each layer decreasing approximately exponentially with the distance from the surface. It is noted that an asymptotic flow pattern around the interface emerges rapidly with the decrease of the length of the bridge.