Unified classification of pure metric geometries

Abstract

Almost Norden, almost product Riemannian, almost Norden golden and almost golden Riemannian are pure metric geometries. We introduce $\alpha$-metric and $\alpha$-golden metric manifolds to unify the study of almost Norden manifolds and almost product Riemannian manifolds with null trace and almost Norden golden manifolds and almost golden Riemannian manifolds with null trace respectively. Then we can show the classifications of almost Norden manifolds and almost product Riemannian manifolds with null trace in a unified way. The bijection between $\alpha$-metric and $\alpha$-golden metric manifolds allows us to classify $\alpha$-golden metric manifolds, i.e., we classify almost Norden golden manifolds and almost golden Riemannian manifolds with null trace simultaneously. Finally we characterize every class of the above four kind of pure metric manifolds by means of the first canonical and the well-adapted connections which are two distinguished connections shared by $\alpha$-metric and $\alpha$-golden metric manifolds.

Keywords

• $\alpha$-metric manifold
• $\alpha$-golden metric manifold
• first canonical connection
• well-adapted connection
• classification

References

1. [1] L. Bilen, S. Turanli and A. Gezer, On Kähler-Norden-Codazzi golden structures on pseudo-Riemannian manifolds, Int. J. Geom. Meth. Mod. Phys. 15 (5), 1850080, 2018.
2. [2] M. Crasmareanu and C.E. Hreţcanu, Golden differential geometry, Chaos, Solitons &Fractals 38 (5), 1229-1238, 2008.
3. [3] S. Erdem, On product and golden structures and harmonicity, Turkish J. Math. 42 (2), 444-470, 2018.
4. [4] F. Etayo, A. deFrancisco and R. Santamaría, The Chern connection of a (J2 = ±1)-metric manifold of class G1, Mediterr. J. Math. 15 (4), Art. 157, 2018.
5. [5] F. Etayo, A. deFrancisco and R. Santamaría, There are no genuine Kähler-Codazzi manifolds Int. J. Geom. Meth. Mod. Phys. 17 (3), 2050044, 2020.
6. [6] F. Etayo, A. deFrancisco and Santamaría, Classification of almost Norden golden manifolds, Bull. Malays. Math. Sci. Soc. 43 (6), 3941-3961, 2020.
7. [7] F. Etayo and R. Santamaría, Distinguished connections on (J2 = ±1)-metric manifold, Arch. Math. (Brno). 52 (3), 59-203, 2016.
8. [8] F. Etayo, and R. Santamaría, The well adapted connection of a (J2 = ±1)-metric manifold, RACSAM 111 (2) 355-375, 2017.

9. [9] F. Etayo, and R. Santamaría, Classification of almost Golden Riemannian manifolds with null trace, Mediterr. J. Math. 17 (3), Art. 90, 2020.
10. [10] F. Etayo, R. Santamaría, and A. Upadhyay, On the geometry of almost Golden Riemannian manifolds, Mediterr. J. Math. 14 (5), Art. 187, 2017.
11. [11] G.T. Ganchev and A.V. Borisov, Note on the almost complex manifolds with a Norden metric, C. R. Acad. Bulgare Sci. 39 (5), 31-34, 1986.
12. [12] G. Ganchev and V. Mihova, Canonical connection and the canonical conformal group on an almost complex manifold with B-metric, Ann. Univ. Sofia Fac. Math. Inform. 81 (1), 195-206, 1987.
13. [13] A. Gezer, N. Cengiz and A. Salimov, On integrability of Golden Riemannian structures, Turkish. J. Math 37 (4), 693-703, 2013.
14. [14] A. Gezer and C. Karaman, Golden-Hessian structures, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 86 (1), 41-46, 2016.
15. [15] S.I. Goldberg and K. Yano, Polynomial structures on manifolds, Kodai Math. Sem. Rep 22 199-218, 1970.
16. [16] R. Kumar, G. Gupta and R. Rani, Adapted connections on Kaehler-Norden Golden manifolds and harmonicity, Int. J. Geom. Meth. Mod. Phys. 17 (2), 2050027, 2020.
17. [17] A.M. Naveira, A classification of Riemannian almost-product manifolds Rend. Mat. (7) 3 (3), 577-592, 1983.
18. [18] A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 157 (4), 925-933, 2004.
19. [19] M. Staikova and K. Gribachev, Canonical connections and their conformal invariants on Riemannian almost product manifolds, Serdica, 18 (3-4), 150-161, 1992.
20. [20] S. Zhong, A. Zhao and G. Mu, Structure-preserving connections on almost complex Norden golden manifolds, J. Geom. 110 (3), Art. 55, 2019