  The Wu invariant is a regular homotopy invariant for immersions of oriented
3-manifolds into the 5-space. In this paper, we present new expressions and
vanishing theorems for the invariant, from the viewpoint of almost contact
structures and complex tangents. As an application, we determine the regular
homotopy classes of inclusion maps of the surface singularity links of type ADE
into the 5-sphere.
