Biharmonic Riemannian Submersions from a 3-Dimensional BCV Space

• Published in: The Journal of geometric analysis Vol. 34; no. 2
• Main Authors: Wang, Ze-Ping, Ou, Ye-Lin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2024; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Classification; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Riemann manifold; Biharmonic Riemannian submersions; Biharmonic maps; 58E20; 53C43; 3-dimensional BCV spaces; Harmonic maps; Riemannian submersions
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01501-9

BCV spaces are a family of 3-dimensional Riemannian manifolds which include six of Thurston’s eight geometries. In this paper, we give a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional BCV space by proving that such biharmonic maps exist only in the cases of H 2 × R → R 2 , or SL ~ ( 2 , R ) → R 2 . In each of these two cases, we are able to construct a family of infinitely many proper biharmonic Riemannian submersions. Our results, on one hand, extend a previous result of the authors which gave a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional space form, and, on the other hand, they can be viewed as the dual study of biharmonic surfaces (i.e., biharmonic isometric immersions) in a BCV space studied in some recent literature.