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Given a Riemannian spin\(^c\) manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O’Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler–Einstein manifold or a Riemannian product of a Kähler–Einstein manifold with \(\mathbb R\) (or with the circle \(\mathbb S^1\)).

Rigidity Results for Riemannian Spin $$^c$$ c Manifolds with Foliated Boundary