Golden- And Product-Shaped Hypersurfaces in Real Space Forms

Abstract

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

“…where I is identity on the tangent bundle of M 2 . In [7], Equation (4) was completely solved for the so-called golden-shaped hypersurfaces in real space forms. We notice that the starting point for the main idea of this study is the Pythagorean Theorem in spite of the fact that the Pythagorean-like formula given by Equation (2) is not directly related to the distance between points as in the usual case.…”

“…Case b. K = 0 and M 2 is an open piece of the Clifford torus. Thus, K ext = −1, which does not fulfill Equation (7).…”

Classification of metallic shaped hypersurfaces in real space forms

“…ii) M is the generalized Clifford torus M = S m (r 1 ) × S n−m (r 2 ) with r with λ 1 λ 2 = −1 (see [8] and [3]). …”

A Neutral relation between metallic structure and almost quadratic φ -structure

“…where J is a (1, 1) tensor field on M, I is the identity operator on the Lie algebra Γ(T M ) of vector fields on M and p, q are real numbers. This structure can be also viewed as a generalization of following well known structures : · If p = 0, q = 1, then J is called almost product or almost para complex structure and denoted by F [15], [12], · If p = 0, q = −1, then J is called almost complex structure [17], · If p = 1, q = 1, then J is called golden structure [6], [7], · If p is positive integer and q = −1, then J is called poly-Norden structure [16], · If p = 1, q = −3 2 , then J is called almost complex golden structure [3], · If p and q are positive integers, then J is called metallic structure [11]. If a differentiable manifold endowed with a metallic structure J then the pair (M, J) is called metallic manifold.…”