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Golden GCR-Lightlike Submanifolds of Golden Semi-Riemannian Manifolds ¨ Nergiz (Onen) Poyraz Abstract. We introduce golden GCR-lightlike submanifolds of golden semi-Riemannian manifolds. We investigate several properties of such submanifolds. Moreover, we find some necessary and sufficient conditions for minimal golden GCR-lightlike submanifolds of golden semiRiemannian manifolds. Mathematics Subject Classification. 53C15, 53C40, 53C50. Keywords. Golden semi-Riemannian manifolds, Golden structure, Lightlike submanifolds, Golden GCR-lightlike submanifolds.

1. Introduction It is well known that in case the induced metric on the submanifold of semiRiemannian manifold is degenerate, the study becomes more different from the study of non-degenerate submanifolds. The primary difference between the lightlike submanifolds and non-degenerate submanifolds arises due to the fact in the first case that the normal vector bundle has non-trivial intersection with the tangent vector bundle and moreover in a lightlike hypersurface the normal vector bundle is contained in the tangent vector bundle. The lightlike submanifolds were introduced by Duggal–Bejancu [4] . Later, they were developed by Duggal and S ¸ ahin [9]. Duggal and Bejancu [4] introduced CR-lightlike submanifolds of indefinite Kaehler manifolds. But CR-lightlike submanifolds exclude the complex and totally real submanifolds as subcases. Then, Duggal and S ¸ ahin introduced screen Cauchy–Riemann (SCR)-lightlike submanifolds of indefinite Kaehler manifolds [6]. But there is no inclusion relation between CR and SCR submanifolds, so Duggal and S ¸ ahin introduced a new class called GCR-lightlike submanifolds of indefinite Kaehler manifolds which is an umbrella for all these types of submanifolds [7] and then of indefinite Sasakian manifolds in [8]. These types of submanifolds have been studied in various manifolds by many authors [13,14,16,17]. 0123456789().: V,-vol

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N. Poyraz

MJOM

Manifolds which are determined differential-geometric structures have an important role in differential geometry. Really, almost complex manifolds and almost product manifolds and maps between such manifolds which are given by a (1,1)-tensor field such that the square of P˜ satisfies certain conditions, like P˜ 2 = −I or P˜ 2 = I, have been studied extensively by many authors. As a generalization of almost complex and almost contact structures, Yano introduced the notion of an f −structure which is a (1,1)-tensor ˜ and satisfies the equality f 3 + f = 0 [22]. It has field of constant rank on M been generalized by Goldberg and Yano in [11]. As a particular case of polynomial structure, Crasmareanu and Hret¸canu studied the golden structures and defined golden Riemannian manifold in [3]. They also investigated the geometry of the golden structure on a manifold by using the corresponding almost product structure. In [21], S ¸ ahin and Akyol introduced golden maps between golden Riemannian manifolds and showed that such maps are harmonic maps. G¨ ok, Kele¸s and Kılı¸c studied some characteriz