Abstract
We introduce semi-slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of semi-slant immersions, invariant Riemannian maps, anti-invariant Riemannian maps and slant Riemannian maps. We obtain characterizations, investigate the harmonicity of such maps and find necessary and sufficient conditions for semi-slant Riemannian maps to be totally geodesic. Then we relate the notion of semi-slant Riemannian maps to the notion of pseudo-horizontally weakly conformal maps, which are useful for proving various complex-analytic properties of stable harmonic maps from complex projective space and give many examples of such maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Abraham, J. E. Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (2nd edition). Applied Mathematical Sciences 75, Springer, New York, 1988.
M. A. Aprodu, M. Aprodu: Implicitly defined harmonic PHH submersions. Manuscr. Math. 100 (1999), 103–121.
M. A. Aprodu, M. Aprodu, V. Brînzănescu: A class of harmonic submersions and minimal submanifolds. Int. J. Math. 11 (2000), 1177–1191.
P. Baird, J. C. Wood: Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monographs. New Series 29, Clarendon Press, Oxford University Press, Oxford, 2003.
A. Bejancu: Geometry of CR-Submanifolds. Mathematics and Its Applications (East European Series) 23, D. Reidel Publishing Co., Dordrecht, 1986.
D. Burns, F. Burstall, P. de Bartolomeis, J. Rawnsley: Stability of harmonic maps of Kähler manifolds. J. Differ. Geom. 30 (1989), 579–594.
P. Candelas, G. T. Horowitz, A. Strominger, E. Witten: Vacuum configurations for superstrings. Nuclear Phys. B (electronic only) 258 (1985), 46–74.
B.-Y. Chen: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Dept. of Mathematics, Leuven, 1990.
B.-Y. Chen: Slant immersions. Bull. Aust. Math. Soc. 41 (1990), 135–147.
D. Chinea: Almost contact metric submersions. Rend. Circ. Mat. Palermo (2) 34 (1985), 89–104.
J. J. Eells, J. H. Sampson: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86 (1964), 109–160.
G. Esposito: From spinor geometry to complex general relativity. Int. J. Geom. Methods Mod. Phys. 2 (2005), 675–731.
M. Falcitelli, S. Ianus, A. M. Pastore: Riemannian Submersions and Related Topics. World Scientific, River Edge, New York, 2004.
A. E. Fischer: Riemannian maps between Riemannian manifolds. Mathematical aspects of classical field theory. Proc. of the AMS-IMS-SIAM Joint Summer Research Conf., Seattle, Washington, USA, 1991 (M. J. Gotay et al., eds.). Contemp. Math. 132, American Mathematical Society, Providence, 1992, pp. 331–366.
E. García-Río, D. N. Kupeli: Semi-Riemannian Maps and Their Applications. Mathematics and Its Applications 475, Kluwer Academic Publishers, Dordrecht, 1999.
A. Gray: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16 (1967), 715–737.
S. Ianuş, R. Mazzocco, G. E. Vîlcu: Riemannian submersions from quaternionic manifolds. Acta Appl. Math. 104 (2008), 83–89.
D. E. Lerner, P. D. Sommers eds.: Complex Manifold Techniques in Theoretical Physics. Research Notes in Mathematics 32, Pitman Advanced Publishing Program, San Francisco, 1979.
E. Loubeau, R. Slobodeanu: Eigenvalues of harmonic almost submersions. Geom. Dedicata 145 (2010), 103–126.
J. C. Marrero, J. Rocha: Locally conformal Kähler submersions. Geom. Dedicata 52 (1994), 271–289.
B. O’Neill: The fundamental equations of a submersion. Mich. Math. J. 13 (1966), 459–469.
N. Papaghiuc: Semi-slant submanifolds of a Kaehlerian manifold. An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 40 (1994), 55–61.
B. Şahin: Slant Riemannian maps to Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 10 (2013), Paper No. 1250080, 12 pages.
B. Şahin: Semi-invariant Riemannian maps to Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 8 (2011), 1439–1454.
B. Şahin: Invariant and anti-invariant Riemannian maps to Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 7 (2010), 337–355.
A. J. Tromba: Teichmüller Theory in Riemannian Geometry: Based on Lecture Notes by Jochen Denzler. Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 1992.
B. Watson: Almost Hermitian submersions. J. Differ. Geom. 11 (1976), 147–165.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Park, KS., Şahin, B. Semi-slant Riemannian maps into almost Hermitian manifolds. Czech Math J 64, 1045–1061 (2014). https://doi.org/10.1007/s10587-014-0152-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-014-0152-3