Given an isometric immersion \(f: M^ m\to {\mathbb{R}}^ n\), its generalized Gauss map g in the sense of Chern and Osserman is the map \(g: M\to G_ m({\mathbb{R}}^ n)\), assigning to each point p in M the tangent space at p of M viewed as an m-dimensional plane in \({\mathbb{R}}^ n\). An easy yet vitally important theorem of \textit{E. A. Ruh} and \textit{J. Vilms} [Trans. Am. Math. Soc. 149, 569-573 (1970; Zbl 0199.561)] states that: g is harmonic if and only if M has parallel mean curvature. Besides the map g, another Gauss map \(\nu\) : TM\({}_ 1^{\perp}\to S^{n-1},called\) the spherical Gauss map, plays an important role in the study of immersed submanifolds of \({\mathbb{R}}^ n\). Its definition is as follows: for each pair (p,e(p)) of the unit normal bundle \(TM_ 1^{\perp}\) of M, \(\nu (p,e(p))=e(p)\) viewed as an element of the unit sphere \(S^{n-1}\). In this paper the author proves that the spherical Gauss map \(\nu\), of a given isometric immersion \(f: M^ m\to {\mathbb{R}}^ n\) is harmonic if and only if M has parallel mean curvature.