Do shapes of matter affect space-time distortion? We know heavy objects bend space-time, but does the curvature only depends on the mass? Or different shapes bend space time differently?
 A: Spacetime is curved by the presence of mass in a perticular point and not by the body's center of mass... so the shape of the body affects the overall curvature because it affects the distribution of mass in the spacetime... the magnitude of the curvature is porpotional to the mass present in a point and the overall "shape" of the curvature is related to the mass distribution on the body...
A: The Einstein field equations read,
$$R_{\mu\nu}-\frac12 g_{\mu\nu}R = 8\pi G\, T_{\mu\nu}$$
where $T_{\mu\nu}$ is the stress-energy-momentum tensor. This is not a constant matrix, it is a tensor field which can depend on space-time.
As such, how the matter is distributed will influence the curvature of space-time and hence ultimately the resultant metric. As a particular example, $T_{\mu\nu}$ with a delta function like distribution often give rise to brane line geometries.
A: The shape of the space around a body depends on its geometry. This Image shows the "shape of the space" in 2 dimensions caused by a spherical symmetric mass. In 3 dimension its a sphere around this mass. So the shape of the space is similar to the geometry of the body. From this you can approximately assume e.g. the shape of the space in the vicinity of a cylinder.    
