Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 197-206, 2017


A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken

Received August 25, 2015.  First published February 24, 2017.

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


References:
[1] C.-L. Bejan: 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 1040052 | Zbl 0683.53034
[2] C.-L. Bejan: The existence problem of hyperbolic structures on vector bundles. Publ. Inst. Math., Nouv. Sér. 53 (67) (1993), 133-138. MR 1319766 | Zbl 0796.53029
[3] C.-L. Bejan: Some examples of manifolds with hyperbolic structures. Rend. Mat. Appl. (7) 14 (1994), 557-565. MR 1312817 | Zbl 0818.53041
[4] C.-L. Bejan, S.-L. Druţă-Romaniuc: Harmonic almost complex structures with respect to general natural metrics. Mediterr. J. Math. 11 (2014), 123-136. DOI 10.1007/s00009-013-0302-0 | MR 3160617 | Zbl 1317.53041
[5] C.-L. Bejan, O. Kowalski: On some differential operators on natural Riemann extensions. Ann. Global Anal. Geom. 48 (2015), 171-180. DOI 10.1007/s10455-015-9463-3 | MR 3376878 | Zbl 06477614
[6] E. Calviño-Louzao, E. García-Río, P. Gilkey, R. Vázquez-Lorenzo: The geometry of modified Riemannian extensions. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 (2009), 2023-2040. DOI 10.1098/rspa.2009.0046 | MR 2515628 | Zbl 1186.53056
[7] V. Cruceanu, P. Fortuny, P. M. Gadea: A survey on paracomplex geometry. Rocky Mt. J. Math. 26 (1996), 83-115. DOI 10.1216/rmjm/1181072105 | MR 1386154 | Zbl 0856.53049
[8] E. García-Río, L. Vanhecke, M. E. Vázquez-Abal: Harmonic endomorphism fields. Illinois J. Math. 41 (1997), 23-30. MR 1433184 | Zbl 0880.53032
[9] A. Gezer, L. Bilen, A. Çakmak: Properties of modified Riemann extension. Zh. Mat. Fiz. Anal. Geom. 11 (2015), 159-173. MR 3442843 | Zbl 1329.53046
[10] I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry. Springer, Berlin (1993). DOI 10.1007/978-3-662-02950-3 | MR 1202431 | Zbl 0782.53013
[11] O. Kowalski, M. Sekizawa: On natural Riemann extensions. Publ. Math. Debrecen 78 (2011), 709-721. DOI 10.5486/PMD.2011.4992 | MR 2867212 | Zbl 1240.53051
[12] O. Kowalski, M. Sekizawa: Almost Osserman structures on natural Riemann extensions. Differ. Geom. Appl. 31 (2013), 140-149. DOI 10.1016/difgeo.2012.10.007 | MR 3010084 | Zbl 1277.53016
[13] E. M. Patterson, A. G. Walker: Riemann extensions. Q. J. Math., Oxf. Ser. (2) 3 (1952), 19-28. DOI 10.1093/qmath/3.1.19 | MR 0048131 | Zbl 0048.15603
[14] A. Salimov, A. Gezer, S. Aslanci: On almost complex structures in the cotangent bundle. Turk. J. Math. 35 (2011), 487-492. DOI 10.3906/mat-0901-31 | MR 2867333 | Zbl 1232.53030
[15] M. Sekizawa: On complete lifts of reductive homogeneous spaces and generalized symmetric spaces. Czech. Math. J. 36 (111) (1986), 516-534. MR 0863184 | Zbl 0615.53042
[16] M. Sekizawa: Natural transformations of affine connections on manifolds to metrics on cotangent bundles. Proc. 14th Winter School Srní, Czech, 1986, Suppl. Rend. Circ. Mat. Palermo, Ser. (2) 14 (1987), 129-142. MR 0920851 | Zbl 0635.53012
[17] T. J. Willmore: An Introduction to Differential Geometry. Clarendon Press, Oxford (1959). MR 0159265 | Zbl 0086.14401
[18] K. Yano, S. Ishihara: Tangent and Cotangent Bundles. Differential Geometry. Pure and Applied Mathematics 16, Marcel Dekker, New York (1973). MR 0350650 | Zbl 0262.53024
[19] K. Yano, E. M. Patterson: Vertical and complete lifts from a manifold to its cotangent bundle. J. Math. Soc. Japan 19 (1967), 91-113. DOI 10.2969/jmsj/01910091 | MR 0206868 | Zbl 0149.19002

Affiliations:   Cornelia-Livia Bejan, Department of Mathematics, \Gh. Asachi\ Technical University of Iaşi, Postal address: Seminar Matematic, Universitatea \Al. I. Cuza\ Iaşi, Bd. Carol I no. 11, Iaşi, 700506, Romania, e-mail: bejanliv@yahoo.com; Şemsi Eken, Department of Mathematics, Mersin University, Çiftlikköy Merkez Street, 33343 Yenişehir, Mersin, Turkey, e-mail: semsieken@hotmail.com


 
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