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A Short Survey on Biharmonic Riemannian Submersions

Year 2024, , 259 - 266, 23.04.2024
https://doi.org/10.36890/iejg.1429642

Abstract

The study of biharmonic submanifolds, initiated by B. Y. Chen and G. Y. Jiang independently, has received a great attention in the past 30 years with many important progress (see the reference for some recent books with vast references therein). This note attempts to give a short survey on the study of biharmonic Riemannian submersions which are a dual concept of biharmonic submanifolds (i.e., biharmonic isometric immersions).

References

  • [1] Akyol, M., Ou, Y.-L.: Biharmonic Riemannian submersions, Annali di Mate. Pura ed Appl., 198 2019, 559–570.
  • [2] Baird, P., Fardoun, A., Ouakkas, S.: Conformal and semi-conformal biharmonic maps, Ann. Glob. Anal. Geom., 34 (2008), 403-414.
  • [3] Baird, P., Wood, J. C.: Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monogr. (N.S.) No. 29, Oxford Univ. Press, 2003.
  • [4] Balmu¸s, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds, J. Geom. Phys., 57. (2007), 449–466.
  • [5] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of S3, Internat. J. Math., 12 (2001), 867–876.
  • [6] Chen, B.-Y.: Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (2) (1991), 169–188.
  • [7] Chen, B.-Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces, Memoirs Fac. Sci. Kyushu Univ. Ser. A, Math., 45(1991), 323–347.
  • [8] Eells, J., Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1) (1978), 1–68.
  • [9] Eells, J., Lemaire, L.: Selected topics in harmonic maps, CBMS, 50, Amer. Math. Soc. (1983).
  • [10] Eells, J., Lemaire, L.: Another report on harmonic maps, Bull. London Math. Soc. 20 (5) (1988), 385–524.
  • [11] Ghandour, E., Ou, Y.-L.: Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps, J. Math. Anal. Appl., 464 (2018), 924-938.
  • [12] Jiang, G. Y.: 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math. Ser. A, 7(1986) 389-402.
  • [13] Jiang, G. Y.: Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math. Ser. A, 8 (1987) 376-383.
  • [14] Loubeau, E., Ou, Y.-L.: Biharmonic maps and morphisms from conformal mappings, Tohoku Math J., 62 (1), (2010), 55-73.
  • [15] O’Neill, B.: The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.
  • [16] Oniciuc, C.: Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi Mat (N.S.) 48 (2002), 237–248.
  • [17] Oniciuc, C.: Biharmonic submanifolds in space forms, Habilitation Thesis, (2012), www.researchgate.net, https://doi.org/10.13140/2.1.4980.5605.
  • [18] Ou, Y.-L., Chen, B.-Y.: Biharmonic submanifolds and biharmonic maps in Riemannian geometry, World Scientific, 2020.
  • [19] Urakawa, H.: Harmonic maps and biharmonic maps on principal bundles and warped products, J. Korean Math. Soc., 55(3), (2018), 553-574.
  • [20] Urakawa, H.: Harmonic maps and biharmonic Riemannian submersions, Note di Mate. 39 (1) (2019), 1–23.
  • [21] Wang, Z.-P., Ou, Y.-L.: Biharmonic Riemannian submersions from 3-manifolds, Math. Zeitschrift, 269 (3) (2011), 917-925.
  • [22] Wang, Z.-P., Ou, Y.-L.: Biharmonic Riemannian submersions from a 3-dimensional BCV space, J. Geom. Anal., 34 (63) 2024 to appear.
  • [23] Wang, Z.-P., Ou, Y.-L.: Biharmonic isometric immersions into and biharmonic Riemannian submersions from M2 × R, preprint 2023, arXiv:2302.11545.
  • [24] Wang, Z.-P., Ou, Y.-L., and Q.-L. Liu Harmonic and biharmonic Riemannain submersions from Sol space, preprint 2023, arXiv:2302.11693.
  • [25] Wang, Z.-P., Ou, Y.-L.: Biharmonic isometric immersions into and biharmonic Riemannian submersions from a generalized Berger sphere , preprint 2023, arXiv:2302.11692.
  • [26] Yau, S.T., Schoen, R. Lectures on harmonic maps, International Press Incorporated, Boston 385 Someville Ave, Someville, MA, U.S.A, 1997.
Year 2024, , 259 - 266, 23.04.2024
https://doi.org/10.36890/iejg.1429642

Abstract

References

  • [1] Akyol, M., Ou, Y.-L.: Biharmonic Riemannian submersions, Annali di Mate. Pura ed Appl., 198 2019, 559–570.
  • [2] Baird, P., Fardoun, A., Ouakkas, S.: Conformal and semi-conformal biharmonic maps, Ann. Glob. Anal. Geom., 34 (2008), 403-414.
  • [3] Baird, P., Wood, J. C.: Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monogr. (N.S.) No. 29, Oxford Univ. Press, 2003.
  • [4] Balmu¸s, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds, J. Geom. Phys., 57. (2007), 449–466.
  • [5] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of S3, Internat. J. Math., 12 (2001), 867–876.
  • [6] Chen, B.-Y.: Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (2) (1991), 169–188.
  • [7] Chen, B.-Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces, Memoirs Fac. Sci. Kyushu Univ. Ser. A, Math., 45(1991), 323–347.
  • [8] Eells, J., Lemaire, L.: A report on harmonic maps, Bull. London Math. Soc. 10 (1) (1978), 1–68.
  • [9] Eells, J., Lemaire, L.: Selected topics in harmonic maps, CBMS, 50, Amer. Math. Soc. (1983).
  • [10] Eells, J., Lemaire, L.: Another report on harmonic maps, Bull. London Math. Soc. 20 (5) (1988), 385–524.
  • [11] Ghandour, E., Ou, Y.-L.: Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps, J. Math. Anal. Appl., 464 (2018), 924-938.
  • [12] Jiang, G. Y.: 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math. Ser. A, 7(1986) 389-402.
  • [13] Jiang, G. Y.: Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math. Ser. A, 8 (1987) 376-383.
  • [14] Loubeau, E., Ou, Y.-L.: Biharmonic maps and morphisms from conformal mappings, Tohoku Math J., 62 (1), (2010), 55-73.
  • [15] O’Neill, B.: The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.
  • [16] Oniciuc, C.: Biharmonic maps between Riemannian manifolds, An. Stiint. Univ. Al. I. Cuza Iasi Mat (N.S.) 48 (2002), 237–248.
  • [17] Oniciuc, C.: Biharmonic submanifolds in space forms, Habilitation Thesis, (2012), www.researchgate.net, https://doi.org/10.13140/2.1.4980.5605.
  • [18] Ou, Y.-L., Chen, B.-Y.: Biharmonic submanifolds and biharmonic maps in Riemannian geometry, World Scientific, 2020.
  • [19] Urakawa, H.: Harmonic maps and biharmonic maps on principal bundles and warped products, J. Korean Math. Soc., 55(3), (2018), 553-574.
  • [20] Urakawa, H.: Harmonic maps and biharmonic Riemannian submersions, Note di Mate. 39 (1) (2019), 1–23.
  • [21] Wang, Z.-P., Ou, Y.-L.: Biharmonic Riemannian submersions from 3-manifolds, Math. Zeitschrift, 269 (3) (2011), 917-925.
  • [22] Wang, Z.-P., Ou, Y.-L.: Biharmonic Riemannian submersions from a 3-dimensional BCV space, J. Geom. Anal., 34 (63) 2024 to appear.
  • [23] Wang, Z.-P., Ou, Y.-L.: Biharmonic isometric immersions into and biharmonic Riemannian submersions from M2 × R, preprint 2023, arXiv:2302.11545.
  • [24] Wang, Z.-P., Ou, Y.-L., and Q.-L. Liu Harmonic and biharmonic Riemannain submersions from Sol space, preprint 2023, arXiv:2302.11693.
  • [25] Wang, Z.-P., Ou, Y.-L.: Biharmonic isometric immersions into and biharmonic Riemannian submersions from a generalized Berger sphere , preprint 2023, arXiv:2302.11692.
  • [26] Yau, S.T., Schoen, R. Lectures on harmonic maps, International Press Incorporated, Boston 385 Someville Ave, Someville, MA, U.S.A, 1997.
There are 26 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Ye-lin Ou 0009-0003-4548-3376

Early Pub Date April 12, 2024
Publication Date April 23, 2024
Submission Date February 1, 2024
Acceptance Date April 1, 2024
Published in Issue Year 2024

Cite

APA Ou, Y.-l. (2024). A Short Survey on Biharmonic Riemannian Submersions. International Electronic Journal of Geometry, 17(1), 259-266. https://doi.org/10.36890/iejg.1429642
AMA Ou Yl. A Short Survey on Biharmonic Riemannian Submersions. Int. Electron. J. Geom. April 2024;17(1):259-266. doi:10.36890/iejg.1429642
Chicago Ou, Ye-lin. “A Short Survey on Biharmonic Riemannian Submersions”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 259-66. https://doi.org/10.36890/iejg.1429642.
EndNote Ou Y-l (April 1, 2024) A Short Survey on Biharmonic Riemannian Submersions. International Electronic Journal of Geometry 17 1 259–266.
IEEE Y.-l. Ou, “A Short Survey on Biharmonic Riemannian Submersions”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 259–266, 2024, doi: 10.36890/iejg.1429642.
ISNAD Ou, Ye-lin. “A Short Survey on Biharmonic Riemannian Submersions”. International Electronic Journal of Geometry 17/1 (April 2024), 259-266. https://doi.org/10.36890/iejg.1429642.
JAMA Ou Y-l. A Short Survey on Biharmonic Riemannian Submersions. Int. Electron. J. Geom. 2024;17:259–266.
MLA Ou, Ye-lin. “A Short Survey on Biharmonic Riemannian Submersions”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 259-66, doi:10.36890/iejg.1429642.
Vancouver Ou Y-l. A Short Survey on Biharmonic Riemannian Submersions. Int. Electron. J. Geom. 2024;17(1):259-66.