On a Study of the Totally Umbilical Semi-Invariant Submanifolds of Golden Riemannian Manifolds

Yıl 2018, Cilt: 21 Sayı: 4, 967 - 970, 01.12.2018

Feyza Esra Erdoğan Cumali Yıldırım

https://doi.org/10.2339/politeknik.389629

Cited By: 11

Öz

The Golden Ratio is fascinating topic that
continually generated news ideas. A Riemannian manifold endowed with a Golden
Structure will be called a Golden Riemannian manifold. Precisely, we can say
that an (1,1)-tensor field  



















 on a m-dimensional Riemann manifold

  is a Golden structure if it satisfies the
equation

,  where

 is identity map on  

.
Furthermore,

,
the Riemannian metric is called   

-compatible
and

  is named a Golden Riemannian manifold. The
main purpose of the present paper is to study the geometry of Riemannian
manifolds endowed with Golden structures.  For
this purpose, we study totally umbilical semi-invariant submanifold of the
Golden Riemannian manifolds. Also, we obtain integrability conditions of the
distributions and investigate the geometry of foliations.

Anahtar Kelimeler

Semi-invariant submanifolds, Golden Riemannian manifolds, totally umbilical submanifolds

Kaynakça

• [1] Bejancu, A., “Geometry of CR-submanifolds”, D. Reidel, (1986). [2] Chen, B.-Y. “Riemannian submanifolds, in Handbook of Differential Geometry”, North Holland (eds. F. Dillen and L. Verstraelen) I, 187418, (2000).
• [3] Crasmareanu, M. and Hretcanu, C. E. “Golden differansiyel geometry”. Chaos, Solitons & Fractals 38, (5): 1229-1238, (2008).
• [4] Crasmareanu, M. and Hretcanu, C. E. “Applications of the Golden Ratio on Riemannian Manifolds”. Turkish J. Math. 33, (2): 179-191, (2009).
• [5] Gezer, A., Cengiz, N. and Salimov, A., “On integrability of Golden Riemannian structures”. Turkish J. Math. 37: 693-703, (2013).
• [6] Hretcanu, C. E and Crasmareanu, M. “On some invariant submanifolds in a Riemannian manifold with golden structure”. An.Stiins. Univ. Al. I. Cuza Iasi. Mat. (N.S.)53, (1): 199-211, (2007).
• [7] Sahin, B. and Akyol, M. A., “Golden maps between Golden Riemannian manifolds and constancy of certain maps”, Math. Commun. 19: 333-342, (2014).
• [8] Yano, K. And Kon, M., “Structures on manifolds”, World Scientific, Singapuore, Series in pure mathematics, 3 (1984).
• [9] Erdoğan, F. E. And Yıldırım C. "Semi-invariant submanifolds of Golden Riemannian manifolds" AIP Conference Proceedings, 1833, 020044, doi:10.1063/1.4981692, (2017).