Abstract
A new concept, namely, D-general warping $$(M=M_1\times M_2,g)$$ , 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 $$\eta $$ -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.
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