Taiwanese Journal of Mathematics

Anti-invariant Riemannian Submersions: A Lie-theoretical Approach

Peter Gilkey, Mitsuhiro Itoh, and JeongHyeong Park

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We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.

Article information

Taiwanese J. Math., Volume 20, Number 4 (2016), 787-800.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 53B20: Local Riemannian geometry 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Riemannian submersion anti-invariant almost Hermitian anti-invariant quaternion anti-invariant para-Hermitian anti-invariant para-quaternion anti-invariant octonian


Gilkey, Peter; Itoh, Mitsuhiro; Park, JeongHyeong. Anti-invariant Riemannian Submersions: A Lie-theoretical Approach. Taiwanese J. Math. 20 (2016), no. 4, 787--800. doi:10.11650/tjm.20.2016.6898. https://projecteuclid.org/euclid.twjm/1498874491

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  • D. V. Alekseevsky and S. Marchiafava, Almost complex submanifolds of quaternionic manifolds, Steps in differential geometry (Debrecen, 2000), 23–28, Inst. Math. Inform., Debrecen, 2001.
  • S. Ali and T. Fatima, Anti-invariant Riemannian submersions from nearly Kaehler manifolds, Filomat 27 (2013), no. 7, 1219–1235.
  • ––––, Generic Riemannian submersions, Tamkang J. Math. 44 (2013), no. 4, 395–409.
  • A. Beri, I. Küpeli Erken and C. Murathan, Anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds, Turkish J. Math. 40 (2016), 540–552.
  • P. Gilkey, J. Park and R. Vázquez-Lorenzo, Aspects of Differential Geometry II, Synthesis Lectures on Mathematics and Statistics 16, Morgan & Claypool, Williston, VT, 2015.
  • Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, J. Funct. Spaces Appl. 2013, Art. ID 720623, 7 pp.
  • N. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), no. 1, 1–55.
  • S. Ivanov and S. Zamkovoy, Parahermitian and paraquaternionic manifolds, Differential Geom. Appl. 23 (2005), no. 2, 205–234.
  • J. Lee, J. Park, B. Sahin and D.-Y. Song, Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut submersions, Taiwanese J. Math. 19, no. 4, 1145–1160.
  • C. Murathan and I. Küpeli Erken, Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds, Filomat 29 (2015), no. 7, 1429–1444.
  • A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), no. 3, 391–404.
  • B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), no. 4, 458–469.
  • J. H. Park, The Laplace-Beltrami operator and Rimanian submersion with minimal and not totally geodesic fibers, Bull. Korean Math. Soc. 27 (1990), 39–47.
  • K. Park, H-anti-invariant submersions from almost quaternionic Hermitian manifolds, arXiv:1507.04473v1 [math.DG].
  • B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, Cent. Eur. J. Math. 8 (2010), no. 3, 437–447.
  • ––––, Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math. 17 (2013), no. 2, 629–659.
  • Wikipedia, https://en.wikipedia.org/wiki/octonian.