From the author's abstract: We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of \(\mathbb{R}^n\) of dimension greater than two? We call an \(n\)-immersion \(f(x)\) in \(\mathbb{R}^m\) isothermic\(_k\) if the normal bundle of \(f\) is flat and \(x\) is a line of curvature coordinate system such that its induced metric is of the form \(\sum_{i=1}^{n} g_{ii}\,\text{d}x_{i}^{2}\) with \(\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0\) , where \(\epsilon_i =1\) for \(1\leq i\leq n-k\) and \(\epsilon_i =-1\) for \(n-k