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Keywords:
Weyl manifold; Einstein-Weyl structure; infinitesimal harmonic transformation
Weyl manifold; Einstein-Weyl structure; infinitesimal harmonic transformation
Summary:
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.
References:
[1] Besse, A. L.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10 Springer, Berlin (1987), German. MR 0867684 | Zbl 0613.53001
[2] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics 203 Birkhäuser, Boston (2010). MR 2682326 | Zbl 1246.53001
[3] Boyer, C. P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs Oxford University Press, Oxford (2008). MR 2382957 | Zbl 1155.53002
[4] Boyer, C. P., Galicki, K., Matzeu, P.: 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] Gauduchon, P.: Weil-Einstein structures, twistor spaces and manifolds of type {$S^1\times S^3$}. J. Reine Angew. Math. 469 (1995), 1-50 French. MR 1363825
[6] Gauduchon, P.: 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] Ghosh, A.: 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] Ghosh, A.: 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] Higa, T.: Weyl manifolds and Einstein-Weyl manifolds. Comment. Math. Univ. St. Pauli 42 (1993), 143-160. MR 1241295 | Zbl 0811.53045
[10] Ishihara, S.: On infinitesimal concircular transformations. Kōdai Math. Semin. Rep. 12 (1960), 45-56. DOI 10.2996/kmj/1138844260 | MR 0121744 | Zbl 0101.14203
[11] Narita, F.: Einstein-Weyl structures on almost contact metric manifolds. Tsukuba J. Math. 22 (1998), 87-98. DOI 10.21099/tkbjm/1496163471 | MR 1637656 | Zbl 0995.53035
[12] Nouhaud, O.: Déformations infinitésimales harmoniques remarquables. C. R. Acad. Sci. Paris Sér. A 275 (1972), 1103-1105 French. MR 0464090 | Zbl 0243.53023
[13] Okumura, M.: 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] Pedersen, H., Swann, A.: Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature. J. Reine Angew. Math. 441 (1993), 99-113. MR 1228613 | Zbl 0776.53027
[15] Pedersen, H., Swann, A.: 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] Perrone, D.: 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] Stepanov, S. E., Shandra, I. G.: Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24 (2003), 291-299. DOI 10.1023/A:1024753028255 | MR 1996772 | Zbl 1035.53090
[18] Stepanov, S. E., Tsyganok, I. I., Mikeš, J.: From infinitesimal harmonic transformations to Ricci solitons. Math. Bohem. 138 (2013), 25-36. MR 3076218 | Zbl 1274.53096
[19] Tanno, S.: The topology of contact Riemannian manifolds. Ill. J. Math. 12 (1968), 700-717. DOI 10.1215/ijm/1256053971 | MR 0234486 | Zbl 0165.24703
[20] Tod, K. P.: 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] Yano, K.: Integral Formulas in Riemannian Geometry. Pure and Applied Mathematics, No. 1 Marcel Dekker, New York (1970). MR 0284950 | Zbl 0213.23801
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