On a Quadratic Functional of the φ -Scalar Curvature
Results in Mathematics, vol.78, no.6, 2023 (SCI-Expanded)
- Publication Type: Article / Article
- Volume: 78 Issue: 6
- Publication Date: 2023
- Doi Number: 10.1007/s00025-023-01983-7
- Journal Name: Results in Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
- Keywords: >φ-scalar flatness, Einstein-type structure, Euler-Lagrange equations, Quadratic functional, φ-scalar curvature
- Istanbul University Affiliated: Yes
Abstract
Given a smooth map φ: M→ N between two Riemannian manifolds (M, g) and (N,〈,〉N) , the φ -scalar curvature of the manifold M, denoted by Sφ , is defined as the trace, with respect to the metric g, of the φ -Ricci tensor, denoted by Ricφ , introduced in [3–5]. In this paper, we focus on the simplest quadratic functional of the φ -scalar curvature Sφ 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 -scalar flat when it is of at least dimension 5 and infMSφ>-∞ . We point out that this result is new also in the special case that φ is constant, that is, in the usual setting of Riemannian geometry.