Kenmotsu manifolds and spin-c Killing spinors

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with an imaginary Killing function $i\mu$ if and only if it is an exact $\mu$-Kenmotsu manifold, thereby obtaining an extension of a recent result by the first named author that, under the purity assumption, does not require simple connectivity or completeness. We then reinterpret $\mu$-Kenmotsu manifolds in terms of metric connections with vectorial torsion and give a second proof of this characterization, combining the theory of complex spinorial forms with the theory of metric connections with torsion. Finally, we describe the global structure of exact $\mu$-Kenmotsu manifolds by means of Morse-Bott theory.