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Abstract
For a Golden-structure ζ on a smooth manifold M and any covariant functor which assigns to M its bundle MA of infinitely near points of A-king, we define the Golden structure ζ^A on M^A and prove that ζ is integrable if and only if so is ζ^A. We also investigate the integrability, parallelism, half parallelism and anti-half parallelism of the Golden-structure ζ^A and their associated distributions on M^A.
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References
- Crˆa¸smˇareanu, M., and Hret¸canu, C.E., Golden differential geometry, Chaos Solitons Fractals, 38:5 (2008), 1229-1238.
- Cruceanu, V., On almost biproduct complex manifolds, An. St. Univ. Al. I. Cuza, Iasi, Mat., 52:1 (2016), 5-24.
- Gezer, A., Cengiz, A., and Salimov, A., On integrability of Golden Riemannian structures, Turkish J. Math., 37 (2013), 693-703.
- G¨ok, M., Kele¸s, S., and Kili¸c, E, Schouten and Vrˇanceanu Connections on Golden Manifolds, Int. Electron. J. Geom., 12:2 (2019), 169-181.
- G¨ok, M., Kele¸s, S., and Kili¸c, E., Some characterisations of semi-invariant submanifolds of golden Reimannian manifolds, Mathematics, 7 (2019), 1-12.
- G¨ok, M., and Kili¸c, E,S., Invariant submanifolds in golden Reimannian manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 125-138.
- G¨ok, M., Kili¸c, E., and Kele¸s, S., Anti-invariant Submanifolds of locally decomposable golden Riemannian manifolds, Balk. J. Geom. Appl., 25 (2020), 47-60.
- G¨ok, M., and Kili¸c, E., Non-invariant submanifolds of locally decomposable golden Riemannian manifolds, Arabian Journal of Mathematics, 10 (2021), 77-89.
- Goldberg, S.I., and Yano, K., Polynomial structures on manifolds, Kodai Math. Sem. Rep., 22 (1970), 199-218.
- Hret¸canu, C., Submanifolds in Riemannian manifold with Golden Structure, Work- shop on Finsler geometry and its applications, Hungary, 2007.
- Hret¸canu, C.E., and Crˆa¸smˇareanu, M., Applications of the golden ratio on Riemannian manifolds, Turkish J. Math., 33 (2009), 179-191.
- Hret¸canu, C.E., and Crˆa¸smˇareanu, M., On some invariant submanifolds in Reimannian manifold with golden structure, An. S¸tiint¸. Univ. A1. I. Cuza Ia¸si. Mat. (N.S.), 53 (2007), 199-211.
- Kol´aˇr, I., Slov´ak, J., and Michor, P.W., Natural operations in differential geometry. SpringerVerlag Berlin Heidelberg, 1993.
- Morimoto, A., Prolongation of connections to bundles of infinitely near points, J. Differ. Geom., 11 (1976), 479-498.
- Nkou, B.V., Bossoto, B.G.R., and Okassa, E., New properties of prolongations of linear connections on weil bundles, Acta Math. Univ. Comenianae, 85:1 (2016), 69-80.
- Ozkan, M., Prolongations of golden structures to tangent bundles, Differential GeometryDynamical Systems, 16 (2014), 227-238.
- Ozkan, M., Prolongations of golden structures to tangent bundles of order r, Commun. Fac. Sci. Univ. Ank. Sr. A1 Math. Stat., 65:1 (2016), 35-47.
- Sahin, B., and Akyol, M.A., Golden maps between Golden Riemannian manifolds and constancy of certain maps, Math. Commun., 19 (2014), 1-10.
- Weil, A., Th´eorie des points proches sur les vari´et´es diff´erentiables. Topologie et G´eom´etrie Diff´erentielle, Colloque du CNRS, Strasbourg, 1953.
- Yano, K., and Kon, M., Structures on manifolds , Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984
References
Crˆa¸smˇareanu, M., and Hret¸canu, C.E., Golden differential geometry, Chaos Solitons Fractals, 38:5 (2008), 1229-1238.
Cruceanu, V., On almost biproduct complex manifolds, An. St. Univ. Al. I. Cuza, Iasi, Mat., 52:1 (2016), 5-24.
Gezer, A., Cengiz, A., and Salimov, A., On integrability of Golden Riemannian structures, Turkish J. Math., 37 (2013), 693-703.
G¨ok, M., Kele¸s, S., and Kili¸c, E, Schouten and Vrˇanceanu Connections on Golden Manifolds, Int. Electron. J. Geom., 12:2 (2019), 169-181.
G¨ok, M., Kele¸s, S., and Kili¸c, E., Some characterisations of semi-invariant submanifolds of golden Reimannian manifolds, Mathematics, 7 (2019), 1-12.
G¨ok, M., and Kili¸c, E,S., Invariant submanifolds in golden Reimannian manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69 (2020), 125-138.
G¨ok, M., Kili¸c, E., and Kele¸s, S., Anti-invariant Submanifolds of locally decomposable golden Riemannian manifolds, Balk. J. Geom. Appl., 25 (2020), 47-60.
G¨ok, M., and Kili¸c, E., Non-invariant submanifolds of locally decomposable golden Riemannian manifolds, Arabian Journal of Mathematics, 10 (2021), 77-89.
Goldberg, S.I., and Yano, K., Polynomial structures on manifolds, Kodai Math. Sem. Rep., 22 (1970), 199-218.
Hret¸canu, C., Submanifolds in Riemannian manifold with Golden Structure, Work- shop on Finsler geometry and its applications, Hungary, 2007.
Hret¸canu, C.E., and Crˆa¸smˇareanu, M., Applications of the golden ratio on Riemannian manifolds, Turkish J. Math., 33 (2009), 179-191.
Hret¸canu, C.E., and Crˆa¸smˇareanu, M., On some invariant submanifolds in Reimannian manifold with golden structure, An. S¸tiint¸. Univ. A1. I. Cuza Ia¸si. Mat. (N.S.), 53 (2007), 199-211.
Kol´aˇr, I., Slov´ak, J., and Michor, P.W., Natural operations in differential geometry. SpringerVerlag Berlin Heidelberg, 1993.
Morimoto, A., Prolongation of connections to bundles of infinitely near points, J. Differ. Geom., 11 (1976), 479-498.
Nkou, B.V., Bossoto, B.G.R., and Okassa, E., New properties of prolongations of linear connections on weil bundles, Acta Math. Univ. Comenianae, 85:1 (2016), 69-80.
Ozkan, M., Prolongations of golden structures to tangent bundles, Differential GeometryDynamical Systems, 16 (2014), 227-238.
Ozkan, M., Prolongations of golden structures to tangent bundles of order r, Commun. Fac. Sci. Univ. Ank. Sr. A1 Math. Stat., 65:1 (2016), 35-47.
Sahin, B., and Akyol, M.A., Golden maps between Golden Riemannian manifolds and constancy of certain maps, Math. Commun., 19 (2014), 1-10.
Weil, A., Th´eorie des points proches sur les vari´et´es diff´erentiables. Topologie et G´eom´etrie Diff´erentielle, Colloque du CNRS, Strasbourg, 1953.
Yano, K., and Kon, M., Structures on manifolds , Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984