Golden differential geometry☆

Prolongations of golden structures to tangent bundles of order r

“…By using Proposition 18 and from ( [2], Theorem 5.1), we get the following theorem. wherer (r) is r lift of a linear connectionr and Q (r) is r lift of an (1; 2) tensor …eld Q for which O P Q is a related Obata operator O P Q (X; Y ) = 1 2 [Q (X; Y ) + P Q (X; P Y )] for the corresponding almost product structure (2.3), then (r) is parallel with respect to r (r) linear connection, i.e.…”

“…Hretcanu [6] studied the golden structure on a manifold M in 2007. Then, Hretcanu and Crasmareanu [2] introduced the geometry of the golden structure on a manifold M by using a corresponding almost product structure. Golden structures were studied by various authors [3,7,8,12,15,16].…”

“…Now we de…ne the operators k and s as follows [2]: k = 1 2 (I + P ) ; s = 1 2 (I P ) where P is an almost product structure.…”

“…Then we get k + s = I; ks = sk = 0; k 2 = k; s 2 = s: (2.5) Equation (2.5) shows that there exist two complementary distributions K and S in M corresponding to the projection operators k and s. k and s are operators providing following relations [2]:…”

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

“…have rich geometric properties. In this way, CR-submanifolds have been studied by many authors see: [2], [3], [7] and they found many intersting results. The structure of this article is following: In Preliminaries, we give some fundamental concepts and definitions needed for this paper.…”