Certain aspects of twisted linear actions. II (Q809421)

Let G be a closed subgroup of \(GL(n,{\mathbb{R}})\) and let \(M=(m_{ij})\) be a square matrix of degree n with \(gM=Mg\) for \(g\in G\), called a G- endomorphism, such that the quadratic form \[ x\cdot Mx=\sum_{i,j}m_{ij}x_ ix_ j \] is positive definite for each \(x\in {\mathbb{R}}^ n-\{0\}\), where \(x=(x_ i)\). For such an M, the author defines an analytic G-action on the sphere \(S^{n-1}\), called a twisted linear action of G on \(S^{n-1}\) determined by the G-endomorphism M, as follows: \[ G\times S^{n-1}\to S^{n-1},\quad (g,x)\mapsto \exp (\tau (gx)M)gx, \] where \(\tau\) is a unique real valued analytic function on \({\mathbb{R}}^ n-\{0\}\) such that \(\| \exp (\tau (x)M)x\| =1\) for \(x\in {\mathbb{R}}^ n-\{0\}.\) In previous papers [ibid. 39, 61-69 (1987; Zbl 0622.57028), Osaka J. Math. 25, No.2, 343-352 (1988; Zbl 0705.57021)], the author has shown that there are uncountably many topologically distinct twisted linear actions of SL(n,\({\mathbb{R}})\) on \(S^{nk-1}\) on \(S^{nk-1}\) for each \(n>k\geq 2\), and there are uncountably many \(C^ 1\)-differentiably distinct but topologically equivalent twisted linear actions of \(SL(n,{\mathbb{R}})\) on \(S^ k\) for each \(k\geq n\geq 2.\) In the present paper, the author shows that there are uncountably many \(C^ 2\)-differentiably distinct but \(C^ 1\)-differentiably equivalent twisted linear actions of \({\mathbb{R}}^ n\) on \(S^ n\) for each \(n\geq 1\). More specifically, for \(n=1\), G (identified with \({\mathbb{R}}^ 1)\) is the closed subgroup of \(GL(2,{\mathbb{R}})\) consisting of matrices of the form \[ \begin{pmatrix} 1&x \\ 0&1 \end{pmatrix};\quad x\in {\mathbb{R}}, \] and the required G-endomorphisms M(a) are of the form \[ \begin{pmatrix} 1&a \\ 0&1 \end{pmatrix};\quad | a| <2. \] Let \(S^ 1(a)\) be the circle \(S^ 1\) with the twisted linear G-action determined by the G-endomorphism M(a). The author proves that for two real numbers a and b with \(| a| <2\) and \(| b| <2\), there exists an equivariant \(C^ 1\)- diffeomorphism from \(S^ 1(a)\) to \(S^ 1(b)\), and there is no equivariant \(C^ 1\)-diffeomorphism from \(S^ 1(a)\) to \(S^ 1(b)\) provided \(a\neq b\). For \(n\geq 2\), the author provides two generalizations of this example, and following Shinichi Watanabe, he concludes his paper studying weakly G-equivariant analytic diffeomorphisms between spheres to the effect that there exist weakly G-equivariant analytic diffeomorphisms between the spheres occurring in the provided examples.