Konuralp Journal of Mathematics, 8 (1) (2020) 38-49 Konuralp Journal of Mathematics Journal Homepage: www.dergipark.gov.tr/konuralpjournalmath e-ISSN: 2147-625X Anti-Invariant Semi-Riemannian Submersions from Indeﬁnite Almost Contact Metric Manifolds Mohammed Danish Siddiqi 1* and Mobin Ahmad 2 1 Department of Mathematics, Faculy of Science Jazan University, Jazan Kingdom of Saudi Arabia 2 Department of Mathematics Faculty of Applied Sciences, Integral University, Lucknow, 226026, U. P., India * Corresponding author Abstract In this paper, we study an anti-invariant semi-Riemmannian submersions from indeﬁnite almost contact metric manifolds. We obtain, the necessary and sufﬁcient conditions for the characteristics vector ﬁled to be vertical and horizontal. aMoreover, we ﬁnd the conditions of integrability and hormonicness of this submersion map. Finally, we furnish an example of an anti-invariant semi-Riemannian submersion from indeﬁnite almost contact metric manifold which is indeﬁnite trans-Sasakian manifolds in the present paper. Keywords: semi-Riemannian submersion; Anti-invaraint submersion; indeﬁnite trans-Sasakian manifolds. 2010 Mathematics Subject Classiﬁcation: 53C15; 53C43; 53B20. 1. Introduction In 1966, the theory of semi-Riemannian submersions between semi-Riemannian manifolds was introduced by O’Neill [3, 4] and Gray [1] in 1967. Watson [2] study Riemannian submersions between almost Hermitian submersions. It is well known that Riemannian submersions are related with physics and have their applications in Kaluza-Klein theory ([14, 25, 26]) Yang-Mills theory ([2, 13]) the theory of supergravity and superstring theories [26]. Afterwords, Sahin introduced anti-invariant and semi-invariant Riemmanin submersion from almost Hermitian manifolds onto Riemannian manifolds. (see [5, 6, 7, 19]). Also, anti-invariant Riemannian submersions extensively studied by several authors (see [16, 17, 28]). In [8], Chinea deﬁned almost contact Riemannian submersion between almost contact metric manifolds. In [12], Lee studied the vertical and horizontal distribution are φ -invariant. Moreover, the characteristic vector ﬁeld ξ is horizontal. We note that only φ -holomorphic submersions have been consider on an almost contact manifolds [21]. Note that notion of anti-invariant submersions was generalized the notion conformal anti-invariant submersions [16]. In fact, anti-invariant Riemannian and Lagrangian submersions have been studying in different kinds of structures such as (see [11, 16, 17]). Recently, in 2018, Siddiqi and Akyol study the some properties of anti-invariant ξ ⊥ -submersions from almost hyperbolic contact manifolds [15, 18]. In [20] Fagahfouri and Mashmouli study anti-invariant semi-Riemannian submersions. In 1980, Oubina [23] introduced the notion of an indeﬁnite trans-Sasakian manifold, of type (α , β ) [10] with indeﬁnite metric play signiﬁcant role in Physics. Indeﬁnite Sasakian manifold is an important kind of indeﬁnite trans-Sasakian manifold with α = 1 and β = 1. Indeﬁnite cosymplectic manifold is another kind of indeﬁnite trans-Sasakian manifold such that α = β = 0. Therefore, motivated by the above studies in this paper, we studied anti-invariant semi-Riemannian submersions from indeﬁnite trans-Sasakian manifolds. 2. Semi-Riemannian submersion In this section, we give necessary background for Semi-Riemannian submersions [9]. Let (M, g) and (N, g N ) be semi-Riemannian manifolds, where dim(M) > dim(N). A surjective map π : (M, g) → (N, g N ) is called a semi-Riemannian submersion [3] if: (S1) π has maximal rank, and Email addresses: msiddiqi@jazanu.edu.sa (Mohammed Danish Siddiqi), mobinahmad@rediffmail.com (Mobin Ahmad)