Araştırma Makalesi
Yıl 2024,
Cilt: 17 Sayı: 1, 259 - 266, 23.04.2024
Öz
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).
Anahtar Kelimeler
Biharmonic maps biharmonic Riemannian submersions BCV spaces Thurston
Kaynakça
- [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.
Yıl 2024,
Cilt: 17 Sayı: 1, 259 - 266, 23.04.2024
Öz
Kaynakça
- [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.
Toplam 26 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebirsel ve Diferansiyel Geometri |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Erken Görünüm Tarihi | 12 Nisan 2024 |
| Yayımlanma Tarihi | 23 Nisan 2024 |
| Gönderilme Tarihi | 1 Şubat 2024 |
| Kabul Tarihi | 1 Nisan 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 17 Sayı: 1 |
