Laplace, Einstein and Related Equations on D-General Warping

A new concept, namely, D-general warping (M=M1xM2,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 TM, 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 -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.