On G-extensible regularity condition and Thom-Boardman singularities
Shyūichi Izumiya · Journal of the Mathematical Society of Japan · 1981
In [2], we have defined a G-extensible reg\={u}larity condition on eq\={u}ivariant sections of differentiable G-fibre b\={u}ndle $P$ .In this paper, we only consider the case where $P$ is a trivial G-fibre b\={u}ndle as an application of Theorem 1.3 in [2].We now form\={u}late as follows:(X, Y)$ be a nat\={u}ral stable reg\={u}larity condition.We say that $\Omega(X, Y)$ is G-extensible if the following conditions hold: There exists a nat\={u}ral stable reg\={u}larity condition $\Omega^{\prime}(X\times R, Y)\subset J^{r}(X\times R, Y)$ (whereFrom [2], we have the following theorem.THEOREM 0.2.Let $C_{G\Omega}^{\infty}(X, Y)$ be the space of the $\Omega$ -regular equivariant maps $X\rightarrow Y$ , with the $C^{\infty}-topology$ , and let $\Gamma_{G}^{0}(\Omega_{G}(X, Y))$ be the space of continuous equiva riant sections of the map $\Omega(X, Y)\cap J_{G}^{r}(X, Y)\rightarrow X$ (with the compact-open topology).Then, if $\Omega(X, Y)$ is G-extensible,