A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan; Şemsi Eken

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 197-206
  • ISSN: 0011-4642

Abstract

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

How to cite

top

Bejan, Cornelia-Livia, and Eken, Şemsi. "A characterization of the Riemann extension in terms of harmonicity." Czechoslovak Mathematical Journal 67.1 (2017): 197-206. <http://eudml.org/doc/287883>.

@article{Bejan2017,
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. 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).},
author = {Bejan, Cornelia-Livia, Eken, Şemsi},
journal = {Czechoslovak Mathematical Journal},
keywords = {semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field},
language = {eng},
number = {1},
pages = {197-206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of the Riemann extension in terms of harmonicity},
url = {http://eudml.org/doc/287883},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Bejan, Cornelia-Livia
AU - Eken, Şemsi
TI - A characterization of the Riemann extension in terms of harmonicity
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 197
EP - 206
AB - 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).
LA - eng
KW - semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field
UR - http://eudml.org/doc/287883
ER -

References

top
  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. (1989) Zbl0683.53034MR1040052
  2. Bejan, C.-L., The existence problem of hyperbolic structures on vector bundles, Publ. Inst. Math., Nouv. Sér. 53 (67) (1993), 133-138. (1993) Zbl0796.53029MR1319766
  3. Bejan, C.-L., Some examples of manifolds with hyperbolic structures, Rend. Mat. Appl. (7) 14 (1994), 557-565. (1994) Zbl0818.53041MR1312817
  4. Bejan, C.-L., Druţă-Romaniuc, S.-L., 10.1007/s00009-013-0302-0, Mediterr. J. Math. 11 (2014), 123-136. (2014) Zbl1317.53041MR3160617DOI10.1007/s00009-013-0302-0
  5. Bejan, C.-L., Kowalski, O., 10.1007/s10455-015-9463-3, Ann. Global Anal. Geom. 48 (2015), 171-180. (2015) Zbl06477614MR3376878DOI10.1007/s10455-015-9463-3
  6. Calviño-Louzao, E., García-Río, E., Gilkey, P., Vázquez-Lorenzo, R., 10.1098/rspa.2009.0046, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 (2009), 2023-2040. (2009) Zbl1186.53056MR2515628DOI10.1098/rspa.2009.0046
  7. Cruceanu, V., Fortuny, P., Gadea, P. M., 10.1216/rmjm/1181072105, Rocky Mt. J. Math. 26 (1996), 83-115. (1996) Zbl0856.53049MR1386154DOI10.1216/rmjm/1181072105
  8. García-Río, E., Vanhecke, L., Vázquez-Abal, M. E., 10.1215/ijm/1255985842, Illinois J. Math. 41 (1997), 23-30. (1997) Zbl0880.53032MR1433184DOI10.1215/ijm/1255985842
  9. Gezer, A., Bilen, L., Çakmak, A., 10.15407/mag11.02.159, Zh. Mat. Fiz. Anal. Geom. 11 (2015), 159-173. (2015) Zbl1329.53046MR3442843DOI10.15407/mag11.02.159
  10. Kolář, I., Michor, P. W., Slovák, J., 10.1007/978-3-662-02950-3, Springer, Berlin (1993). (1993) Zbl0782.53013MR1202431DOI10.1007/978-3-662-02950-3
  11. Kowalski, O., Sekizawa, M., 10.5486/PMD.2011.4992, Publ. Math. Debrecen 78 (2011), 709-721. (2011) Zbl1240.53051MR2867212DOI10.5486/PMD.2011.4992
  12. Kowalski, O., Sekizawa, M., 10.1016/difgeo.2012.10.007, Differ. Geom. Appl. 31 (2013), 140-149. (2013) Zbl1277.53016MR3010084DOI10.1016/difgeo.2012.10.007
  13. Patterson, E. M., Walker, A. G., 10.1093/qmath/3.1.19, Q. J. Math., Oxf. Ser. (2) 3 (1952), 19-28. (1952) Zbl0048.15603MR0048131DOI10.1093/qmath/3.1.19
  14. Salimov, A., Gezer, A., Aslancı, S., 10.3906/mat-0901-31, Turk. J. Math. 35 (2011), 487-492. (2011) Zbl1232.53030MR2867333DOI10.3906/mat-0901-31
  15. Sekizawa, M., On complete lifts of reductive homogeneous spaces and generalized symmetric spaces, Czech. Math. J. 36 (111) (1986), 516-534. (1986) Zbl0615.53042MR0863184
  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) (1987), 129-142. (1987) Zbl0635.53012MR0920851
  17. Willmore, T. J., An Introduction to Differential Geometry, Clarendon Press, Oxford (1959). (1959) Zbl0086.14401MR0159265
  18. Yano, K., Ishihara, S., Tangent and Cotangent Bundles. Differential Geometry, Pure and Applied Mathematics 16, Marcel Dekker, New York (1973). (1973) Zbl0262.53024MR0350650
  19. Yano, K., Patterson, E. M., 10.2969/jmsj/01910091, J. Math. Soc. Japan 19 (1967), 91-113. (1967) Zbl0149.19002MR0206868DOI10.2969/jmsj/01910091

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.