A characterization of the Riemann extension in terms of harmonicity

• Published in: Czechoslovak mathematical journal Vol. 67; no. 1; pp. 197 - 206
• Main Authors: Bejan, Cornelia-Livia, Eken, Şemsi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017; Springer Nature B.V
• Subjects: Analysis; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; cotangent bundle; harmonic tensor field; 53C50; 58E20; 53C43; semi-Riemannian manifold; 53B05; natural Riemann extension; 53C07
• ISSN: 0011-4642; 1572-9141
• DOI: 10.21136/CMJ.2017.0459-15

If ( M ,∇) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure P on ( T * M , g ¯ ) and prove that P is harmonic (in the sense of E.Garciá-Río, L.Vanhecke and M. E.Vázquez-Abal (1997)) if and only if g ¯ reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952).