If is purely imaginary, then is a
noncompact manifold; if is 0, then the spinor field is parallel; finally, if is real, then is compact, and the spinor field is called a ``real spinor field."
References
^Th. Friedrich (1980). "Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung". Mathematische Nachrichten. 97: 117–146.
doi:
10.1002/mana.19800970111.
^Th. Friedrich (1989). "On the conformal relation between twistors and Killing spinors". Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II. 22: 59–75.