Araştırma Makalesi
Yıl 2025,
Cilt: 54 Sayı: 1, 75 - 89, 28.02.2025
Öz
Kaynakça
- [1] M. T. K. Abbassi and M. Sarih, On some hereditary properties of Riemannian gnatural metrics on tangent bundles of Riemannian manifolds, Difer. Geom. Appl. 22, 19–47, 2005.
- [2] R. Albuquerque, The ciconia metric on the tangent bundle of an almost Hermitian manifold, Ann. Mat. Pura Appl. 199 (3), 969–984, 2020.
- [3] M. Altunbas, R. Simsek and A. Gezer, Some harmonic problems on the tangent bundle with a Berger-type deformed Sasaki metric, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2), 37–42, 2020.
- [4] P. Baird and C. L. Bejan, Quasi-harmonic maps between almost symplectic manifolds, Chapman & amp; Hall/CRC Res. Notes Math., 413 Chapman & Hall/CRC, Boca Raton, FL, 2000, 75–95.
- [5] P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monograph (New Series), Oxford University Press, New York (2003).
- [6] A. Balmus, C. Oniciuc and N. Papaghiuc, Harmonic properties on the tangent bundle, An. Univ. Vest Timisoara, Ser. Mat.-Inform 42 , 17–28, 2004.
- [7] C. L. Bejan, Quasi-harmonic morphisms, Ital. J. Pure Appl. Math. 18 , 51– 68, 2005.
- [8] C. L. Bejan and S. L. Druta-Romaniuc, Connections which are harmonic with respect to general natural metrics, Diff. Geom. Appl. 30 (4), 306–317, 2012.
- [9] C. L. Bejan and S. L. Druta-Romaniuc, Harmonic almost complex structures with respect to general natural metrics, Mediterr. J. Math. 11 (1), 123–136, 2014.
- [10] A. Bejancu and H. Faran, Foliations and Geometric Structures, Math. Appl. 580, Springer, Dordrecht, 2006.
- [11] A. Bejancu and H. Faran, Vrãnceanu connections and foliations with bundle-like metrics, Proc. Indian Acad. Sci. Math. Sci. 118 (1), 99113, 2008.
- [12] M. Benyounes, E. Loubeau and C. M.Wood, Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type, Differ. Geom. Appl. 25 (3), 322–334, 2007.
- [13] B. Y. Chen and T. Nagano, Harmonic metrics, harmonic tensors and Gauss maps, J. Math. Soc. Jpn. 36, 295–313, 1984.
- [14] C. T. J. Dodson, P. M. Trinidad and M. E. Vazquez-Abal, Harmonic-Killing vector fields, Bull. Belg. Math. Soc. Simon Stevin 9 (4), 481–490, 2002.
- [15] J. Eells and L. Lemaire, Selected topics in harmonic maps, Conf. Board of the Math.Sci. A.M.S. 50, 85 pp., 1983.
- [16] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109–160, 1964.
- [17] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic connections, Acta Sci. Math. (Szeged) 62 (3–4), 583–607, 1996.
- [18] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic endomorphism fields, J. Math. 41 (1), 23–30, 1997.
- [19] A. Gezer, L. Bilen and U. C. De, Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric, Filomat 37 (24), 8193–8204, 2023.
- [20] S. Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ 13, 23–27, 1979.
- [21] O. Nouhaud, Applications harmoniques d’une variete Riemanniene dans son fibre tangent, C. R. Acad. Sci. Paris Sér. 284 (14), 815–818, 1997.
- [22] Z. Olszak, The Schouten Van-Kampen affine connection adapted to an almost(para) contact metric structure, Publications De l’institut Mathématique, 94 (114), 31–42, 2013.
- [23] C. Oniciuc, Tangency and Harmonicity Properties, Ph.D. Thesis, DGDS Monographs 3, Geometry Balkan Press, Bucuresti, 2003.
- [24] V. Oproiu, On the harmonic sections of cotangent bundles, Rend. Sem. Fac. Sci. Univ. Cagliari 59 (2), 177–184, 1989.
- [25] M. P. Piu, Campi di vettori ed applicazioni armoniche, Rend. Sem. Fac. Sci. Univ. Cagliari 52, 85–94, 1982.
- [26] A. A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 154 (4), 925-933, 2007.
- [27] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (3), 338-354, 1958.
- [28] J. A. Schouten and E. R. Van Kampen, Zur Einbettungs-und Krümmungstheorie nichtholonomer Gebilde, Math. Ann. 103, 752-783, 1930.
- [29] M. Spivak, A Comprehensive Introduction to differential geometry vol 2, Publish or perish, INC Houston, Texas 1999.
- [30] G. Vrãnceanu, Sur quelques points de la théories des espaces non holonomomes, Bull. Fac. St. Cernãuti 5, 177–205, 1931.
- [31] K. Yano and M. Ako, On certain operators associated with tensor field, Kodai Math. Sem. Rep. 20, 414-436, 1968.
- [32] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc. New York, USA, 1973.
Yıl 2025,
Cilt: 54 Sayı: 1, 75 - 89, 28.02.2025
Öz
The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection $\pi :\left( TM,\tilde{g}\right) \rightarrow \left( M_{2n},J,g\right) $, where $\left( M_{2n},J,g\right) $ represents an anti-paraK\"{a}hler manifold and $\left( TM,\tilde{g}\right) $ its tangent bundle with the ciconia metric. Additionally, we study the harmonicity of a vector field $\xi$, treated as mappings from $M$ to $TM$ . In this context, we consider the harmonicity relations between the ciconia metric $\tilde{g}$ and the Sasaki metric $^{S}g$, examining their mutual interactions. Furthermore, we investigate the Schoutan-Van Kampen connection and the Vr\~{a}nceanu connection, both associated with the Levi-Civita connection of the ciconia metric. Our analysis also includes the computation of the mean connections for the Schoutan-Van Kampen and Vr\~{a}nceanu connections, thereby providing insights into their properties. Finally, our exploration extends to the second fundamental form of the identity mapping from $\left( TM,\tilde{g}\right) $ to $\left(TM,\overline{\nabla }^{m}\right) ~$ and $\left( TM,\widetilde{\nabla }^{\ast m}\right) $. Here $\overline{\nabla }^{m}$ and $\widetilde{\nabla }^{\ast m}$ denote the mean connections associated with the Schoutan-Van Kampen and Vr\~{a}nceanu connections, respectively.
Anahtar Kelimeler
Kaynakça
- [1] M. T. K. Abbassi and M. Sarih, On some hereditary properties of Riemannian gnatural metrics on tangent bundles of Riemannian manifolds, Difer. Geom. Appl. 22, 19–47, 2005.
- [2] R. Albuquerque, The ciconia metric on the tangent bundle of an almost Hermitian manifold, Ann. Mat. Pura Appl. 199 (3), 969–984, 2020.
- [3] M. Altunbas, R. Simsek and A. Gezer, Some harmonic problems on the tangent bundle with a Berger-type deformed Sasaki metric, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 82 (2), 37–42, 2020.
- [4] P. Baird and C. L. Bejan, Quasi-harmonic maps between almost symplectic manifolds, Chapman & amp; Hall/CRC Res. Notes Math., 413 Chapman & Hall/CRC, Boca Raton, FL, 2000, 75–95.
- [5] P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monograph (New Series), Oxford University Press, New York (2003).
- [6] A. Balmus, C. Oniciuc and N. Papaghiuc, Harmonic properties on the tangent bundle, An. Univ. Vest Timisoara, Ser. Mat.-Inform 42 , 17–28, 2004.
- [7] C. L. Bejan, Quasi-harmonic morphisms, Ital. J. Pure Appl. Math. 18 , 51– 68, 2005.
- [8] C. L. Bejan and S. L. Druta-Romaniuc, Connections which are harmonic with respect to general natural metrics, Diff. Geom. Appl. 30 (4), 306–317, 2012.
- [9] C. L. Bejan and S. L. Druta-Romaniuc, Harmonic almost complex structures with respect to general natural metrics, Mediterr. J. Math. 11 (1), 123–136, 2014.
- [10] A. Bejancu and H. Faran, Foliations and Geometric Structures, Math. Appl. 580, Springer, Dordrecht, 2006.
- [11] A. Bejancu and H. Faran, Vrãnceanu connections and foliations with bundle-like metrics, Proc. Indian Acad. Sci. Math. Sci. 118 (1), 99113, 2008.
- [12] M. Benyounes, E. Loubeau and C. M.Wood, Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type, Differ. Geom. Appl. 25 (3), 322–334, 2007.
- [13] B. Y. Chen and T. Nagano, Harmonic metrics, harmonic tensors and Gauss maps, J. Math. Soc. Jpn. 36, 295–313, 1984.
- [14] C. T. J. Dodson, P. M. Trinidad and M. E. Vazquez-Abal, Harmonic-Killing vector fields, Bull. Belg. Math. Soc. Simon Stevin 9 (4), 481–490, 2002.
- [15] J. Eells and L. Lemaire, Selected topics in harmonic maps, Conf. Board of the Math.Sci. A.M.S. 50, 85 pp., 1983.
- [16] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109–160, 1964.
- [17] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic connections, Acta Sci. Math. (Szeged) 62 (3–4), 583–607, 1996.
- [18] E. Garcia-Rio, L. Vanhecke and M. E. Vazquez-Abal, Harmonic endomorphism fields, J. Math. 41 (1), 23–30, 1997.
- [19] A. Gezer, L. Bilen and U. C. De, Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric, Filomat 37 (24), 8193–8204, 2023.
- [20] S. Ishihara, Harmonic sections of tangent bundles, J. Math. Tokushima Univ 13, 23–27, 1979.
- [21] O. Nouhaud, Applications harmoniques d’une variete Riemanniene dans son fibre tangent, C. R. Acad. Sci. Paris Sér. 284 (14), 815–818, 1997.
- [22] Z. Olszak, The Schouten Van-Kampen affine connection adapted to an almost(para) contact metric structure, Publications De l’institut Mathématique, 94 (114), 31–42, 2013.
- [23] C. Oniciuc, Tangency and Harmonicity Properties, Ph.D. Thesis, DGDS Monographs 3, Geometry Balkan Press, Bucuresti, 2003.
- [24] V. Oproiu, On the harmonic sections of cotangent bundles, Rend. Sem. Fac. Sci. Univ. Cagliari 59 (2), 177–184, 1989.
- [25] M. P. Piu, Campi di vettori ed applicazioni armoniche, Rend. Sem. Fac. Sci. Univ. Cagliari 52, 85–94, 1982.
- [26] A. A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 154 (4), 925-933, 2007.
- [27] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (3), 338-354, 1958.
- [28] J. A. Schouten and E. R. Van Kampen, Zur Einbettungs-und Krümmungstheorie nichtholonomer Gebilde, Math. Ann. 103, 752-783, 1930.
- [29] M. Spivak, A Comprehensive Introduction to differential geometry vol 2, Publish or perish, INC Houston, Texas 1999.
- [30] G. Vrãnceanu, Sur quelques points de la théories des espaces non holonomomes, Bull. Fac. St. Cernãuti 5, 177–205, 1931.
- [31] K. Yano and M. Ako, On certain operators associated with tensor field, Kodai Math. Sem. Rep. 20, 414-436, 1968.
- [32] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc. New York, USA, 1973.
Toplam 32 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebirsel ve Diferansiyel Geometri |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 14 Nisan 2024 |
| Yayımlanma Tarihi | 28 Şubat 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 54 Sayı: 1 |