DOI: 10.36890/iejg.1855272 ISSN: 1307-5624

Clairaut Semi-Invariant Riemannian Maps from Trans-Sasakian Manifolds

Murat Polat
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We investigate Clairaut semi-invariant Riemannian maps defined on trans-Sasakian manifolds. Starting from a differential approach to the Clairaut condition, we derive an intrinsic characterization showing that the Clairaut property is equivalent to the total umbilicity of the fibers together with a specific relation between the mean curvature vector field and the gradient of the Clairaut function. This characterization is obtained independently of any contact-type structure and thus holds for general semi-invariant Riemannian maps. When the trans-Sasakian structure is taken into account, a clear dichotomy appears according to the position of the Reeb vector field. In the vertical Reeb case, the Clairaut condition exhibits a rigid behavior analogous to the Kähler manifold, with no explicit contribution from the trans-Sasakian structure functions. In contrast, when the Reeb vector field is horizontal, the trans-Sasakian parameter $\alpha$ enters the Clairaut characterization and produces additional terms in the mean curvature and curvature formulas. We further analyze consequences for total geodesicity of the fibers, harmonicity of the map, and curvature behavior, showing that the trans-Sasakian effects are visible precisely in the mixed and Ricci curvature components. Several explicit examples illustrate that the two cases lead to genuinely different geometric behaviors and confirm the sharpness of the obtained results.

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