Laplace, Einstein and Related Equations on D-General Warping

• Published in: Mediterranean journal of mathematics Vol. 16; no. 1; pp. 1 - 21
• Main Authors: Bejan, Cornelia-Livia, Güler, Sinem
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2019; Springer Nature B.V
• Subjects: Curvature; Geometry; Mathematics; Mathematics and Statistics; Metric space; Warping; Secondary 53C15; quasi-Einstein; Riemann extension; 58J05; almost contact metric; 53C04; warping; Primary 53C25; Harmonicity
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-018-1283-9

A new concept, namely, D -general warping ( M = M 1 × M 2 , g ) , is introduced by extending some geometric notions defined by Blair and Tanno. Corresponding to a result of Tanno in almost contact metric geometry, an outcome in almost Hermitian context is provided here. On T ∗ M , the Riemann extension (introduced by Patterson and Walker) of the Levi–Civita connection on ( M ,  g ) is characterized. A Laplacian formula of g is obtained and the harmonicity of functions and forms on ( M ,  g ) is described. Some necessary and sufficient conditions for ( M ,  g ) to be Einstein, quasi-Einstein or η -Einstein are provided. The cases when the scalar (resp. sectional) curvature is positive or negative are investigated and an example is constructed. Some properties of ( M ,  g ) for being a gradient Ricci soliton are considered. In addition, D -general warpings which are space forms (resp. of quasi-constant sectional curvature in the sense of Boju, Popescu) are studied.