Geodesic
Properties of Modified Riemannian Extensions
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2015) no. 2, pp. 159-173 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $M$ be an $n$-dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast }M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $\widetilde{g}_{\nabla ,c}$ on $T^{\ast }M$ defined by means of a symmetric $(0,2)$-tensor field $c$ on $M.$ We get the conditions under which $T^{\ast }M $ endowed with the horizontal lift $^{H}J$ of an almost complex structure $J$ and with the metric $\widetilde{g}_{\nabla ,c}$ is a Kähler–Norden manifold. Also curvature properties of the Levi–Civita connection of the metric $\widetilde{g}_{\nabla ,c}$ are presented. 
@article{JMAG_2015_11_2_a2,
     author = {A. Gezer and L. Bilen and A. Cakmak},
     title = {Properties of {Modified} {Riemannian} {Extensions}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {159--173},
     year = {2015},
     volume = {11},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2015_11_2_a2/}
}
TY  - JOUR
AU  - A. Gezer
AU  - L. Bilen
AU  - A. Cakmak
TI  - Properties of Modified Riemannian Extensions
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2015
SP  - 159
EP  - 173
VL  - 11
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JMAG_2015_11_2_a2/
LA  - en
ID  - JMAG_2015_11_2_a2
ER  - 
%0 Journal Article
%A A. Gezer
%A L. Bilen
%A A. Cakmak
%T Properties of Modified Riemannian Extensions
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2015
%P 159-173
%V 11
%N 2
%U http://geodesic.mathdoc.fr/item/JMAG_2015_11_2_a2/
%G en
%F JMAG_2015_11_2_a2
A. Gezer; L. Bilen; A. Cakmak. Properties of Modified Riemannian Extensions. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2015) no. 2, pp. 159-173. http://geodesic.mathdoc.fr/item/JMAG_2015_11_2_a2/

[1] S. Aslanci, S. Kazimova, A. A. Salimov, “Some Remarks Concerning Riemannian Extensions”, Ukrainian Math. J., 62:5 (2010), 661–675 | DOI

[2] Z. Afifi, “Riemann Extensions of Affine Connected Spaces”, Quart. J. Math., Oxford Ser., 5 (1954), 312–320 | DOI

[3] E. Calvino-Louzao, E. García-Río, P. Gilkey, A. Vazquez-Lorenzo, “The Geometry of Modified Riemannian Extensions”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465:2107 (2009), 2023–2040 | DOI

[4] E. Calviño-Louzao, E. García-Río, R. Vázquez-Lorenzo, “Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor”, Can. J. Math., 62:5 (2010), 1037–1057 | DOI

[5] A. Derdzinski, “Connections with Skew-Symmetric Ricci Tensor on Surfaces”, Results Math., 52:3–4 (2008), 223–245 | DOI

[6] L. S. Druţă, “Classes of General Natural Almost Anti-Hermitian Structures on the Cotangent Bundles”, Mediterr. J. Math., 8:2 (2011), 161–179 | DOI

[7] V. Dryuma, “The Riemann Extensions in Theory of Differential Equations and their Applications”, Mat. Fiz., Anal., Geom., 10:3 (2003), 307–325

[8] E. Garcia-Rio, D. N. Kupeli, M. E. Vazquez-Abal, R. Vazquez-Lorenzo, “Affine Osserman Connections and their Riemann Extensions”, Diff. Geom. Appl., 11:2 (1999), 145–153 | DOI

[9] T. Ikawa, K. Honda, “On Riemann Extension”, Tensor (N.S.), 60:2 (1998), 208–212

[10] M. Iscan, A. A. Salimov, “On Kähler–Norden Manifolds”, Proc. Indian Acad. Sci. Math. Sci., 119:1 (2009), 71–80 | DOI

[11] O. Kowalski, M. Sekizawa, “On Natural Riemann Extensions”, Publ. Math. Debrecen, 78:3–4 (2011), 709–721 | DOI

[12] K. P. Mok, “Metrics and Connections on the Cotangent Bundle”, Kodai Math. Sem. Rep., 28:2–3 (1976/77), 226–238

[13] V. Oproiu, N. Papaghiuc, “On the Cotangent Bundle of a Differentiable Manifold”, Publ. Math. Debrecen, 50:3–4 (1997), 317–338

[14] V. Oproiu, N. Papaghiuc, “Some Examples of Aalmost Complex Manifolds with Norden Metric”, Publ. Math. Debrecen, 41:3–4 (1992), 199–211

[15] E. M. Patterson, A. G. Walker, “Riemann Extensions”, Quart. J. Math. Oxford Ser., 3 (1952), 19–28 | DOI

[16] A. Salimov, “On Operators Associated with Tensor Fields”, J. Geom., 99:1–2 (2010), 107–145 | DOI

[17] M. Sekizawa, “Natural Transformations of Affine Connections on Manifolds to Metrics on Cotangent Bundles”, Proceedings of the 14th winter school on abstract analysis (Srni, 1986), Rend. Circ. Mat. Palermo (2), no. 14, 1987, 129–142

[18] Z. I. Szabo, “Structure Theorems on Riemannian Spaces Satisfying $R(X,Y)R=0$. I: The Local Version”, J. Differential Geom., 17 (1982), 531–582

[19] M. Toomanian, Riemann Extensions and Complete Lifts of $s$-spaces, Ph. D. Thesis, University of Southampton, 1975

[20] S. Tachibana, “Analytic Tensor and its Generalization”, Tohoku Math. J., 12:2 (1960), 208–221 | DOI

[21] L. Vanhecke, T. J. Willmore, “Riemann Extensions of D'Atri Spaces”, Tensor (N.S.), 38 (1982), 154–158

[22] T. J. Willmore, “Riemann Extensions and Affine Differential Geometry”, Results Math., 13:3–4 (1988), 403–408 | DOI

[23] K. Yano, M. Ako, “On Certain Operators Associated with Tensor Field”, Kodai Math. Sem. Rep., 20 (1968), 414–436 | DOI

[24] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc., New York, 1973