Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 31 (2021), No. 2, 393--411Copyright Heldermann Verlag 2021 The Liouville Theorem of a Torsion System and its Application to the Symmetry Group of a Porous Medium Type Equation on Symmetric Spaces Xiao-Peng Chen Dept. of Mathematics, Shantou University, Shantou 515063, P. R. China xpchen@stu.edu.cn Shi-Zhong Du Dept. of Mathematics, Shantou University, Shantou 515063, P. R. China szdu@stu.edu.cn Tian-Pei Guo Dept. of Mathematics, Shantou University, Shantou 515063, P. R. China 18tpguo@stu.edu.cn [Abstract-pdf] We first prove a Liouville theorem to the torsion system $$\begin{cases} \displaystyle \xi^i_i=\lambda(x)\pm\frac{2x^k\xi^k}{|x|^2+1}, & \forall i=1,2,\cdots,n\\ \xi^i_j+\xi^j_i=0, & \forall i\not=j \end{cases}$$ for $(\xi,\lambda)\in C^\infty({\mathbb{R}}^n,{\mathbb{R}}^n\times{\mathbb{R}})$. As an application, complete resolutions of symmetry groups to the porous medium equation $$u_t-\triangle_g(u^m)=u^p, \ \ \forall(x,t)\in M\times{\mathbb{R}}$$ of Fujita type are obtained, where $M$ is the sphere ${\mathbb{S}}^n\subset{\mathbb{R}}^{n+1}$ or hyperbolic space ${\mathbb{H}}^n$ with canonical metric $g$. Keywords: Porous medium equation, prolongation formula. MSC: 53C35, 35K59, 35K65. [ Fulltext-pdf  (150  KB)] for subscribers only.