A characterization of the Riemann extension in terms of harmonicity

Bejan, Cornelia-Livia ; Eken, Şemsi

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 197-206

Abstract (VO)

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\bar {g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar {g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $\mathcal {P}$ on $(T^{*}M,\bar {g})$ and prove that $\mathcal {P}$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar {g}$ reduces to the classical Riemann extension introduced by E. M. Patterson and A. G. Walker (1952).

Détail

DOI : 10.21136/CMJ.2017.0459-15

Classification : 53B05, 53C07, 53C43, 53C50, 58E20
Keywords: semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field

Bibliographie

[1] Bejan, C.-L.: A classification of the almost para-Hermitian manifolds. Differential Geometry and Its Applications N. Bokan et al. Proc. of the Conf. Dubrovnik, 1988, Univ. Novi Sad, Inst. of Mathematics, Novi Sad (1989), 23-27. | MR | JFM

[2] Bejan, C.-L.: The existence problem of hyperbolic structures on vector bundles. Publ. Inst. Math., Nouv. Sér. 53 (67) (1993), 133-138. | MR | JFM

[3] Bejan, C.-L.: Some examples of manifolds with hyperbolic structures. Rend. Mat. Appl. (7) 14 (1994), 557-565. | MR | JFM

[4] Bejan, C.-L., Druţă-Romaniuc, S.-L.: Harmonic almost complex structures with respect to general natural metrics. Mediterr. J. Math. 11 (2014), 123-136. | DOI | MR | JFM

[5] Bejan, C.-L., Kowalski, O.: On some differential operators on natural Riemann extensions. Ann. Global Anal. Geom. 48 (2015), 171-180. | DOI | MR | JFM

[6] Calviño-Louzao, E., García-Río, E., Gilkey, P., Vázquez-Lorenzo, R.: The geometry of modified Riemannian extensions. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 (2009), 2023-2040. | DOI | MR | JFM

[7] Cruceanu, V., Fortuny, P., Gadea, P. M.: A survey on paracomplex geometry. Rocky Mt. J. Math. 26 (1996), 83-115. | DOI | MR | JFM

[8] García-Río, E., Vanhecke, L., Vázquez-Abal, M. E.: Harmonic endomorphism fields. Illinois J. Math. 41 (1997), 23-30. | DOI | MR | JFM

[9] Gezer, A., Bilen, L., Çakmak, A.: Properties of modified Riemann extension. Zh. Mat. Fiz. Anal. Geom. 11 (2015), 159-173. | DOI | MR | JFM

[10] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993). | DOI | MR | JFM

[11] Kowalski, O., Sekizawa, M.: On natural Riemann extensions. Publ. Math. Debrecen 78 (2011), 709-721. | DOI | MR | JFM

[12] Kowalski, O., Sekizawa, M.: Almost Osserman structures on natural Riemann extensions. Differ. Geom. Appl. 31 (2013), 140-149. | DOI | MR | JFM

[13] Patterson, E. M., Walker, A. G.: Riemann extensions. Q. J. Math., Oxf. Ser. (2) 3 (1952), 19-28. | DOI | MR | JFM

[14] Salimov, A., Gezer, A., Aslancı, S.: On almost complex structures in the cotangent bundle. Turk. J. Math. 35 (2011), 487-492. | DOI | MR | JFM

[15] Sekizawa, M.: On complete lifts of reductive homogeneous spaces and generalized symmetric spaces. Czech. Math. J. 36 (111) (1986), 516-534. | MR | JFM

[16] Sekizawa, M.: Natural transformations of affine connections on manifolds to metrics on cotangent bundles. Proc. 14th Winter School Srn' ı, Czech, 1986, Suppl. Rend. Circ. Mat. Palermo, Ser. (2) {\it 14} (1987), 129-142. | MR | JFM

[17] Willmore, T. J.: An Introduction to Differential Geometry. Clarendon Press, Oxford (1959). | MR | JFM

[18] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles. Differential Geometry. Pure and Applied Mathematics 16, Marcel Dekker, New York (1973). | MR | JFM

[19] Yano, K., Patterson, E. M.: Vertical and complete lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan 19 (1967), 91-113. | DOI | MR | JFM