On a Quadratic Functional of the $\varphi$-Scalar Curvature

Abstract

Given a smooth map $\varphi :M \rightarrow N$ between two Riemannian manifolds (M, g) and $\big (N,{\left\langle {~},{~}\right\rangle }_{N}\big )$, the $\varphi$-scalar curvature of the manifold M, denoted by $S^{\varphi }$, is defined as the trace, with respect to the metric g, of the $\varphi$-Ricci tensor, denoted by $Ric^{\varphi }$, introduced in [3,4,5]. In this paper, we focus on the simplest quadratic functional of the $\varphi$-scalar curvature $S^{\varphi }$ 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 $\varphi$-scalar flat when it is of at least dimension 5 and $\inf _{M}\, S^{\varphi }>-\infty$. We point out that this result is new also in the special case that $\varphi$ is constant, that is, in the usual setting of Riemannian geometry.