Interaction of Codazzi Pairs with Almost Para Norden Manifolds

Year 2022, , 212 - 227, 30.06.2022

Sibel Turanlı , Sedanur Uçan

https://doi.org/10.47000/tjmcs.1075806

Cited By: 1

Abstract

In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an almost para Norden manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate connection of a linear connection $\mathrm{\nabla }$ on $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\
},\ {\mathrm{\nabla }}^{\dagger \ }$ and ${\mathrm{\nabla }}^{\varphi \ }$ are defined and it is demonstrated that on the spaces of linear connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as the four-element Klein group. We also searched some properties of these three types conjugate
connections. Then, Codazzi pairs $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ and $\left(\mathrm{\nabla },G\right)$ are introduced and some properties of them are given. Let $R\ ,\ R^{*\ }$and $R^{\dagger \ }$are $(0,4)$-curvature tensors of conjugate connections
$\mathrm{\nabla }\mathrm{\ ,\ }{\mathrm{\nabla }}^{*\ }$and ${\mathrm{\nabla }}^{\dagger \ }$, respectively. The relationship among the curvature tensors is investigated. The condition of $N_{\varphi }=0$ is obtained, where $N_{\varphi }$ is Nijenhuis tensor field on $M_{2n}$ and it is known
that the condition of integrability of almost para complex structure $\varphi $ is $N_{\varphi }=0$. In addition, Tachibana operator is applied to the pure metric $g$ and a necessary and sufficient condition $\left(M,\varphi ,\ g,G\right)$ being a para Kahler Norden manifold is found. Finally, we examine $\varphi $-invariant linear connections and statistical manifolds.

Keywords

Codazzi pairs, connections, Norden manifolds, statistical structures

References

• Blaga, M., A. Crasmareanu, M., The geometry of complex conjugate connections, Hacettepe Journal of Mathematics and Statistics, 41(1)(2012), 119-126.
• Blaga, M., A. Crasmareanu, M., The geometry of tangent conjugate connections, Hacettepe Journal of Mathematics and Statistics, 44(4)(2015), 767-774.
• Calın, O., Matsuzoe, H., Zhang, J., Generalizations of conjugate connections, Trends in Differential Geometry, Complex Analysis and Mathematical Physics, (2009), 26-34.
• Catino, G., Mantegazza, C., Mazzieri, L., A note on Codazzi tensors, Math. Ann., 362(2015), 629-638.
• D'atri, J.E., Codazzi tensors and harmonic curvature for left invariant metrics, Geometriae Dedicata, 19(3 (1985), 229-236.
• Derdzinski, A., Shen, C., Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc., 47(3)(1983), 15-26.
• Derdzinski, A., Some Remarks on The Local Structure of Codazzi Tensors, Global Differential Geometry and Global Analysis, Springer, Berlin, Heidelberg, 1981.
• Dillen, F., Nomizu, K., Vranken, L., Conjugate connections and Radon's theorem in affine differential geometry, Mh. Math., 109(1990), 221-235.
• Etayo, F., Santamaria, R., $\left( J^{2}=\mp 1\right) -$ metric manifolds, Publ. Math. Debrecen, 57(3-4)(2000), 435-444.
• Fei, T., Zhang, J., Interaction of Codazzi couplings with (Para-) Kahler geometry, Results Math., 72(2017), 2037-2056.
• Ganchev, G.T., Borisov, A.V., Note on the almost complex manifolds with a Norden metric, C. R. Acad. Bulgarie Sci., 39(5)(1986), 31-34.
• Gebarowski, A., The structure of Certain Riemannian manifolds admitting Codazzi tensors, Demonstratio Mathematica, 27(1)(1994), 249-252.
• Gezer, A., Cakicioglu, H., Notes concerning Codazzi pairs on almost anti-hermitian manifolds, In press.
• İşcan, M., Sarsılmaz, H., Turanli, S., On 4-dimensional almost Para complex pure walker Mmnifolds, Turkish Journal of Mathematics, 38(2014), 1071-1080.
• Kruchkovich, G.I., Hypercomplex structure on a manifold, I. Tr. Sem. Vect. Tens. Anal. Moscow Univ., 16(1972), 174-201.
• Lauritzen, S.L., Statistical Manifolds, In: Differential geometry in statistical inferences, IMS Lecture Notes Monograph Series Institute of Mathematical Statistics, 10(1987), 163-216.
• Nomizu, K., Simon, U., Notes on Conjugate Connections, In Geometry and Topology of Submanifolds IV, eds. F. Dillen and L. Verstraelen, Word Scientific, 1992.
• Pinkall, U., Schwenk-Schellschmidt, A., Simon, U., Geometric methods for solving Codazzi and Monge-Ampere equations, Math. Ann., 298(1994), 89-100.
• Salimov, A.A., Almost analyticity of a Riemannian metric and integrability of a structure, Trudy Geom. Sem. Kazan. Univ., 15(1983), 72-78.
• Salimov, A.A., On operators associated with tensor fields, J. Geom., 99(1-2)(2010), 107-145.
• Salimov, A.A., Iscan, M., Akbulut, K., Notes on Para-Norden-Walker 4-Manifolds, International Journal of Geometric Methods in Modern Physics, 7(8)(2010), 1331-1347.
• Salimov, A.A., Iscan, M., Etayo, F., Paraholomorphic B-manifold and its properties, Topology Appl., 154(2007), 925-933.
• Salimov, A.A., Turanli, S., Curvature properties of anti-Kahler Codazzi manifolds, C.R. Acad. Sci. Paris. Ser. I, 351(5-6)(2013), 225-227.
• Salimov, A.A., Akbulut, K., Turanli, S., On an İsotropic property of anti-Kahler Codazzi manifolds, C.R. Acad. Sci. Paris. Ser. I, 351(21-22)(2013), 837-839.
• Schwenk-Schellschmidt, A., Simon, U., Codazzi-equivalent affine connections, Result. Math., 56(2009), 211-229.
• Shandra, G.I., Stepanov, S.E., Mikes, J., On Higher-order Codazzi tensors on complete Riemannian manifolds, Annals of Global Analysis and Geometry, 56(2019), 429-442.
• Simon, U., A further method in differential geometry, Abh. Math. Sem. Hamburg, 44(1)(1975), 52-69.
• Simon, U., Codazzi Tensors, Global Differential Geometry and Global Analysis, Springer, Berlin, Heidelberg, 1981.
• Simon, U., Schwenk-Schellschmidt, A., Vrancken, L., Codazzi-equivalent Riemannian metrics, Asian J. Math., 14(3)(2010), 291-302.
• Tachibana, A., Analytic tensor and its generalization}, T\^{o}hoku Math. J., 12(2)(1960), 208-221.
• Vishnevskii, V.V., Integrable affinor structures and their plural interpretations, J. of Math. Sciences, 108(2)(2002), 151-187.