Laplace, Einstein and Related Equations on D-General Warping
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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.
KeywordsHarmonicity quasi-Einstein almost contact metric Riemann extension D-warping
Mathematics Subject ClassificationPrimary 53C25 Secondary 53C15 53C04 58J05
Both authors would like to thank all three referees for their reports and for carefully reading the manuscript. The second author thanks The Scientific and Technological Research Council of Turkey (TÜBİTAK) for financial support (Grant Number: 1059B141600696), which helps to carry out a part of this work. The second author is also supported by Research Fund of the Istanbul Technical University (GAP project reference TGA-2018-41211).
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