h-projectively equivalent metrics

• Applicant: Professor Dr. Vladimir Matveev
• Subject Area: Mathematics
• Term: from 2012 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 226608726

• Final Report Year: 2018

Final Report Abstract

C-projectively equivalent metrics and c-projective transformations is a natural concept in the Kähler geometry: is was introduced in the 50th in the context of a natural structures and natural transformations of Kähler manifolds, and then was independently re-invented and re-introduced in the 90th the context of ina tegrable system (under the name “Kähler-Liouville metrics”) and in 2000th as generalization of the so-called Calabi construction (under the name “Hamiltonian 2-forms”). The circles of methods that were used, and also the goals, were also completely different. The main idea of the project was to combine both the methods and the goals. This approach appeared to be very productive. The most important results were the local description of c-projectively equivalent metrics and the proof of the Yano-Obata conjecture, in the Riemannian case on complete manifolds and in the case of any signature on compact manifolds. Both problems were explicitly asked in 1950th-1970th and were one of the main goals of the research in the direction.

Publications

• C-projective geometry, Mem. AMS
  D. Calderbank, M. Eastwood, V. Matveev, K. Neusser
  
• Local description of Bochner-flat (pseudo-)Kähler metrics, Comm. Anal. Geom.
  A. Bolsinov, S. Rosemann
  
• Local normal forms for cprojectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics
  A. Bolsinov, V. Matveev, S. Rosemann
  
• Proof of the Yano-Obata Conjecture for holomorph-projective transformations, J. Diff. Geom. 92 (2012) 221– 261
  V. Matveev, S. Rosemann
  (See online at https://doi.org/10.4310/jdg/1352297807)
• Conification construction for Kähler manifolds and its application in c-projective geometry, Adv. Math. 274 (2015), 1–38
  V. Matveev, S. Rosemann
  (See online at https://doi.org/10.1016/j.aim.2015.01.006)
• Four-dimensional Kähler metrics admitting c-projective vector fields, J. Math. Pures Appl. (9) 103 (2015), no. 3, 619–657
  A. Bolsinov, V. Matveev, Th. Mettler, S. Rosemann
  (See online at https://doi.org/10.1016/j.matpur.2014.07.005)
• Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms, Compositio Math. 152 (2016), 1555–1575
  D. Calderbank, V. Matveev, S. Rosemann
  (See online at https://doi.org/10.1112/S0010437X16007302)
• Submaximal c-projective structures, Int. J. Math. 27 (2016), 1650022
  B. Kruglikov, V. Matveev and D. The
  (See online at https://doi.org/10.1142/S0129167X16500221)
• On the groups of c-projective transformations of complete Kähler manifolds, Ann. Global Anal. Geom. 54(2018), no. 3, 329–352
  V. Matveev, K. Neusser
  (See online at https://doi.org/10.1007/s10455-018-9604-6)
• On the Lichnerowicz conjecture for CR manifolds with mixed signature, Comptes Rendus 356 (2018) 532–537
  J. S. Case, S. N. Curry, V. Matveev
  (See online at https://doi.org/10.1016/j.crma.2018.03.012)