A note on uncountable groups with modular subgroup lattice

In this paper, we obtain a rigidity theorem by modifying Cheng–Yau’s technique to linear Weingarten submanifolds in the unit sphere S^n+p(1) with parallel normalized mean curvature vector. As a corollary, we have Theorem 1.3 in Guo and Li (Tohoku Math J 65:331–339, 2013) and Theorem 2 in Li (Math Ann 305:665–672, 1996).