Pullback theorem and rigidity for Sobolev mappings on Carnot groups

Abstract: We study the pullback theorem of Sobolev mappings on Carnot groups via mollification of mappings. With the pullback theorem we extend the classical result proved by Xiangdong Xie : Rigidity of Sobolev mappings $W^{1,p}(G_1;G_2)$ for $p>ν$, to the case $p<ν$, where $ν$ is the homogeneous dimension of $G_1$. Therefore, some conclusions about continuity of Sobolev mappings on Carnot groups for $p<ν$ are found. And also, the determine of horizontal gradient $D_Hf$ is invariant under the motion related to higher layer left-invariant vector fields. At last, we find a equivalent definition of quasiconformal mappings with lower integrability $dim(g^{[1]})<p<ν$.