reduces the general case to the above one.

0.3cm Both Proposition \[P:covering\] and Corollary \[C:covering\] will be used in the following sections when we consider concrete examples.

0.3cm

The converse of Corollary \[C:covering\] is not true. That is, the isotropic subgroups $G_y$ associated with $y\in Y'$ may vary “suddenly", even if $Y'$ is connected. For example, The group $SO(n)$ acts on $\mathbb{R}P^n$ smoothly if we regard $\mathbb{R}P^n$ as the quotient space by gluing the opposite points on the boundary of the closed unit ball $B^n$. Let $X_{\mathrm{z}}$ be the image of $\{0\}$, $Y$ be the image of the segment $\{(x,0,\cdots,0):|x|\leq 1\}$, then the conditions (a), (b), (c), and (e) hold. The isotropic subgroup associated with the image of a point in $Y'$ which is an interior point of $B^n$ is ${\mathrm{diag}}(1,SO(n-1))$, but for the image of the point $(1,0,\cdots,0)$, its isotropic subgroup is ${\mathrm{diag}}(\pm1,O^\pm(n-1))$ (here $O^\pm(n-1)=\{g\in O(n-1):\det
g=\pm1\}$ ). Other examples with the similar phenomena will appeared in [@AWY] when we consider the group ensemble associated with complex semisimple Lie groups. When the phenomena of sudden variation of the isotropic subgroups happens, whether we can in general make them to be of the same by enlarging the set $X_{\mathrm{z}}$ is 