Given a smooth map φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ between two Riemannian manifolds (M, g) and φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ, the φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ-scalar curvature of the manifold M, denoted by φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ, is defined as the trace, with respect to the metric g, of the φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ-Ricci tensor, denoted by φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ, introduced in [3–5]. In this paper, we focus on the simplest quadratic functional of the φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ-scalar curvature φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ of M and we observe that its Euler-Lagrange equations give rise to a particular Einstein-type structure on M as defined in [3]. With the aid of the latter together with the completeness of g and two more mild assumptions, we are able to conclude that M is φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ-scalar flat when it is of at least dimension 5 and φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ. We point out that this result is new also in the special case that φ:M→N(N,,N)φSφφRicφφSφφinfMSφ>-∞φ is constant, that is, in the usual setting of Riemannian geometry.