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The central idea of this research article is to examine the characteristics of Clairaut submersions from Lorentzian trans-Sasakian manifolds of type (α, β) and also, to enhance this geometrical analysis with some specific cases, namely Clairaut submersion from Lorentzian α-Sasakian manifold, Lorentzian β-Kenmotsu manifold and Lorentzian cosymplectic manifold. Furthermore, the authors discuss some results about Clairaut Lagrangian submersions whose total space is a Lorentzian trans-Sasakian manifolds of type (α, β). Finally, the authors furnished some examples based on this study.

This research discourse based on classifications of submersion, mainly Clairaut submersions, whose total manifolds is Lorentzian trans-Sasakian manifolds and its all classes like Lorentzian Sasakian, Lorenztian Kenmotsu and Lorentzian cosymplectic manifolds. In addition, the authors have explored some axioms of Clairaut Lorentzian submersions and illustrates our findings with some non-trivial examples.

The major finding of this study is to exhibit a necessary and sufficient condition for a submersions to be a Clairaut submersions and also find a condition for Clairaut Lagrangian submersions from Lorentzian trans-Sasakian manifolds.

The results and examples of the present manuscript are original. In addition, more general results with fair value and supportive examples are provided.

The authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly  symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.

The authors have used the tensorial approach to achieve the goal.

A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.

The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.

In this research paper, the authors delve into the ethnography and socioeconomic status of the Chenchu tribal community that resides in Telangana's Nallamalla forest. The authors also examine the initiatives undertaken by the government to create fair entrepreneurial opportunities while preserving their unique culture. Additionally, the authors analyse potential entrepreneurial and business opportunities using the environmental, social and governance (ESG) framework.

To gain insight into the current socio-economic development and living standards of the Chenchu tribal people, the authors conducted interviews in nine different villages with 177 individual respondents. Statistical analysis was then used to determine the relationship between business opportunities and their impact on socio-economic development based on the data collected.

The Chenchu tribal community has faced significant challenges, including vulnerability, poverty and debt. To promote the development of the tribe, it is essential to provide equal opportunities in education, healthcare and means of subsistence. The government has implemented various social and developmental programs to support sustainable development and entrepreneurial opportunities while preserving the tribe's unique culture and improving their education and standard of living. These initiatives are aimed at benefiting the entire community.

The Chenchu tribal people are eager for progress but hesitant to leave their forest homes and integrate into modern society. This creates challenges for efficiently implementing welfare programs. It is crucial to strike a balance between forest regulations and the rights of indigenous communities to ensure their protection and prevent any hindrance to development efforts.

To promote sustainable development and preserve tribal cultures, the government should implement various social and developmental programs. These programs should improve living conditions and socioeconomic development while exploring business opportunities. Maintaining these communities' unique identities while striving for progress is essential.