Surface velocities in parts of the India-Asia collision zone are compared to velocities calculated from equations describing fluid flow driven by topographically produced pressure gradients. A good agreement is found if the viscosity of the crust is ~1020 Pa s in southern Tibet and ~1022 Pa s in the area between the Eastern Syntaxis and the Szechwan Basin. The lower boundary condition of the flow changes between these two areas, with a stress-free lower boundary in the area between the Szechwan basin and the Eastern Syntaxis, and a horizontally rigid but vertically deformable boundary where strong Indian lithospheric material underlies southern Tibet. Deformation maps for olivine, diopside and anorthite show our findings to be consistent with laboratory measurements of the rheology of minerals. Gravitationally driven flow is also suggested to be taking place in the Indo-Burman Ranges, with a viscosity of ~1019-1020 Pa s. Flow in both southern Tibet and the Indo-Burman Ranges provides an explanation for the formation of the geometry of the Eastern Himalayan Syntaxis. The majority of the normal faulting earthquakes in the Tibetan Plateau occur in the area of southern Tibet which we model as gravitationally spreading over the Indian shield.