In this paper we employ the
techniques introduced by Wu-Yi Hsiang in [4] to perform Stiefel-Whitney class
calculations for the possibilities of connected principal isotropy type listed in
Theorems 1–3 of [4]. We show that some of the possibilities listed there do
not occur if we assume in addition that sufficiently many Stiefel-Whitney
classes of the G-manifold vanish. We therefore obtain a slightly shorter list of
possibilities of connected principal isotropy type for compact connected Lie group
actions on parallelizable manifolds. Stiefel manifolds which are not spheres,
for example, fall under this category. We also give an example of how our
results may be used to study actions on Stiefel manifolds. As this paper is
actually a supplement to [4], we refer the reader to it for notation and general
philosophy.
