Classification of Almost Norden Golden Manifolds
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Classification of Almost Norden Golden Manifolds Fernando Etayo1
· Araceli deFrancisco2
· Rafael Santamaría2
Received: 14 October 2019 / Revised: 15 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Almost Norden manifolds were classified by means of the Levi–Civita connection by Ganchev and Borisov and by means of the canonical connection by Ganchev and Mihova. This canonical connection was obtained using the potential tensor of the Levi–Civita of the twin metric. We recall that this canonical connection is the welladapted connection, obtaining its explicit expression for some classes and being able to obtain two equivalent classifications of almost Norden golden manifolds. Keywords Almost Norden manifold · Almost Norden golden manifold · Well-adapted connection · Classification Mathematics Subject Classification 53C15 · 53C05 · 53C07
1 Introduction Almost complex and almost golden structures on a manifold are polynomial structures of degree 2, i.e., structures given by a tensor field of type (1, 1) satisfying a polynomial equation of degree 2 (see [12]).
Communicated by Young Jin Suh.
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Fernando Etayo [email protected] Araceli deFrancisco [email protected] Rafael Santamaría [email protected]
1
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39071 Santander, Spain
2
Departamento de Matemáticas, Escuela de Ingenierías Industrial, Informática y Aeroespacial, Universidad de León, Campus de Vegazana, 24071 León, Spain
123
F. Etayo et al.
Definition 1 Let M be a manifold and let Id be the identity tensor field of type (1, 1) on M. (i) A polynomial structure J of degree 2 on M satisfying J 2 = −Id is called an almost complex structure. In this case, (M, J ) is an almost complex manifold. (ii) A polynomial structure ϕ of degree 2 on M satisfying ϕ 2 = ϕ − 23 Id is called an almost complex golden structure. In this case, (M, ϕ) is an almost complex golden manifold. Almost complex golden structures were introduced by Crasmareanu and Hre¸tcanu in [2]. This kind of structures is the analogue of almost golden structures in the complex case (also introduced in the quoted paper). An almost golden structure on a manifold is also a polynomial structure ϕ of degree satisfying ϕ 2 = ϕ + Id. The characteristic polynomials of almost golden and almost complex golden structures are x 2 − x −1 and √ √ x 2 − x − 23 , respectively, whose roots are φ = 1+2 5 and φ¯ = 1−2 5 in the golden case, √ √ and φc = 1+2 5i and φ¯ c = 1−2 5i in the complex golden case. Recall that the numbers φ and φ¯ c are called the golden ratio and the complex golden ratio, respectively. These numbers explain the name of both kind of polynomial structures. Since then, almost golden and almost complex golden structures have become in active research field (see, e.g., [1,3,6,10,11] and the references therein). Almost complex and almost complex golden structures are closely related as it is shown in two of the above-mentioned refe
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