Pullback theorem and rigidity for Sobolev mappings on Carnot groups

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 $p>\nu$, to the case $p<\nu$, where $\nu$ is the homogeneous dimension of $G_1$. Therefore, some conclusions about continuity of Sobolev mappings on Carnot groups for $p<\nu$ 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 $dim(g^{[1]})<p<\nu$.