On a Quadratic Functional of the φ -Scalar Curvature

Rigoli M., Yıldırım H.

Results in Mathematics, cilt.78, sa.6, 2023 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 78 Sayı: 6
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1007/s00025-023-01983-7
  • Dergi Adı: Results in Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Anahtar Kelimeler: >φ-scalar flatness, Einstein-type structure, Euler-Lagrange equations, Quadratic functional, φ-scalar curvature
  • İstanbul Üniversitesi Adresli: Evet

Özet

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.