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Complete Riemannian manifolds admitting a pair of Einstein-Weyl structures

Amalendu Ghosh

Received October 25, 2014.   First published June 16, 2016.

Abstract: We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pm\omega)$ with constant scalar curvature is either Einstein, or the dual field of $\omega$ is Killing. Next, let $(M^n, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pm\omega)$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $\omega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.

Keywords: Weyl manifold; Einstein-Weyl structure; infinitesimal harmonic transformation

Classification (MSC 2010): 53C25, 53C15, 53C20

DOI: 10.21136/MB.2016.0072-14

Full text available as PDF.

References:
  [1] A. L. Besse: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10 Springer, Berlin (1987), German. DOI 10.1007/978-3-540-74311-8 | MR 0867684 | Zbl 0613.53001
  [2] D. E. Blair: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203 Birkhäuser, Boston (2010). DOI 10.1007/978-0-8176-4959-3 | MR 2682326 | Zbl 05771354
  [3] C. P. Boyer, K. Galicki: Sasakian Geometry. Oxford Mathematical Monographs Oxford University Press, Oxford (2008). MR 2382957 | Zbl 1155.53002
  [4] C. P. Boyer, K. Galicki, P. Matzeu: On $\eta$-Einstein Sasakian geometry. Comm. Math. Phys. 262 (2006), 177-208. DOI 10.1007/s00220-005-1459-6 | MR 2200887 | Zbl 1103.53022
  [5] P. Gauduchon: Weil-Einstein structures, twistor spaces and manifolds of type {$S^1\times S^3$}. J. Reine Angew. Math. 469 (1995), 1-50 French. DOI 10.1515/crll.1995.469.1 | MR 1363825 | Zbl 0858.53039
  [6] P. Gauduchon: La 1-forme de torsion d'une variété hermitienne compacte. Math. Ann. 267 (1984), 495-518 French. DOI 10.1007/BF01455968 | MR 0742896 | Zbl 0523.53059
  [7] A. Ghosh: Certain infinitesimal transformations on contact metric manifolds. J. Geom. 106 (2015), 137-152. DOI 10.1007/s00022-014-0240-4 | MR 3320884 | Zbl 1319.53091
  [8] A. Ghosh: Einstein-Weyl structures on contact metric manifolds. Ann. Global Anal. Geom. 35 (2009), 431-441. DOI 10.1007/s10455-008-9145-5 | MR 2506245 | Zbl 1180.53031
  [9] T. Higa: Weyl manifolds and Einstein-Weyl manifolds. Comment. Math. Univ. St. Pauli 42 (1993), 143-160. MR 1241295 | Zbl 0811.53045
  [10] S. Ishihara: On infinitesimal concircular transformations. Kodai Math. Semin. Rep. 12 (1960), 45-56. DOI 10.2996/kmj/1138844260 | MR 0121744 | Zbl 0101.14203
  [11] F. Narita: Einstein-Weyl structures on almost contact metric manifolds. Tsukuba J. Math. 22 (1998), 87-98. MR 1637656 | Zbl 0995.53035
  [12] O. Nouhaud: Déformations infinitésimales harmoniques remarquables. C. R. Acad. Sci. Paris Sér. A 275 (1972), 1103-1105 French. MR 0464090 | Zbl 0243.53023
  [13] M. Okumura: Some remarks on space with a certain contact structure. Tohoku Math. J. (2) 14 (1962), 135-145. DOI 10.2748/tmj/1178244168 | MR 0143148 | Zbl 0119.37701
  [14] H. Pedersen, A. Swann: Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature. J. Reine Angew. Math. 441 (1993), 99-113. DOI 10.1515/crll.1993.440.99 | MR 1228613 | Zbl 0776.53027
  [15] H. Pedersen, A. Swann: Riemannian submersions, four-manifolds and Einstein-Weyl geometry. Proc. Lond. Math. Soc. (3) 66 (1993), 381-399. DOI 10.1112/plms/s3-66.2.381 | MR 1199072 | Zbl 0742.53014
  [16] D. Perrone: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Differ. Geom. Appl. 20 (2004), 367-378. DOI 10.1016/j.difgeo.2003.12.007 | MR 2053920 | Zbl 1061.53028
  [17] S. E. Stepanov, I. G. Shandra: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24 (2003), 291-299. DOI 10.1023/A:1024753028255 | MR 1996772 | Zbl 1035.53090
  [18] S. E. Stepanov, I. I. Tsyganok, J. Mikeš: From infinitesimal harmonic transformations to Ricci solitons. Math. Bohem. 138 (2013), 25-36. MR 3076218 | Zbl 1274.53096
  [19] S. Tanno: The topology of contact Riemannian manifolds. Ill. J. Math. 12 (1968), 700-717. MR 0234486 | Zbl 0165.24703
  [20] K. P. Tod: Compact 3-dimensional Einstein-Weyl structures. J. Lond. Math. Soc. (2) 45 (1992), 341-351. DOI 10.1112/jlms/s2-45.2.341 | MR 1171560 | Zbl 0761.53026
  [21] K. Yano: Integral Formulas in Riemannian Geometry. Pure and Applied Mathematics, No. 1 Marcel Dekker, New York (1970). MR 0284950 | Zbl 0213.23801