Mediterr. J. Math. (2017) 14:187
published online August 14, 2017
Springer International Publishing AG 2017
On the Geometry of Almost Golden
Fernando Etayo, Rafael Santamar´ıa
and Abhitosh Upadhyay
Abstract. An almost Golden Riemannian structure (ϕ, g) on a manifold
is given by a tensor ﬁeld ϕ of type (1,1) satisfying the Golden section
= ϕ + 1, and a pure Riemannian metric g, i.e., a metric
satisfying g(ϕX, Y )=g(X, ϕY ). We study connections adapted to such
a structure, ﬁnding two of them, the ﬁrst canonical and the well adapted,
which measure the integrability of ϕ and the integrability of the G-
structure corresponding to (ϕ, g).
Mathematics Subject Classiﬁcation. Primary 53C15; Secondary 53C07,
Keywords. Almost Golden structure, pure Riemannian metric, adapted
connection, ﬁrst canonical connection, well-adapted connection.
The Golden section (or Golden mean or Golden ratio) φ has a remarkable
story, coming from ancient times. Deﬁnition of φ can be given as follows: Can
there be a segment divided into two portions, one bigger than the other, in
such a way that the ratio between the whole segment and the big portion is
equal to that of the big and the small portions? If the measure of the small
portion is 1, the measure φ of the big portion must satisfy:
− φ − 1=0
thus proving φ must be a root of the polynomial equation x
− x − 1=0.
In fact, φ =
, because the other root, usually denoted by
=1− φ<0, not corresponding to a ratio of lengths.
The Golden ratio is used in many of the remarkable constructions and
paintings in history, because it expresses an idea of harmony and “good”
proportion. Besides, it also appears in Nature, adding interest and fascination.
From a mathematical point of view, it has been extensively studied in books
such as [8,10]. In the last years the Golden mean can be found in many areas