DOI: 10.1007/s00209-003-0503-5 Math. Z. 244, 337–347 (2003) Mathematische Zeitschrift Extrinsic Killing spinors Oussama Hijazi 1 , Sebasti ´ an Montiel 2 1 Institut ´ Elie Cartan, Universit´ e Henri Poincar´ e, Nancy I, B.P. 239, 54506Vandœuvre- L` es-Nancy Cedex, France (e-mail: hijazi@iecn.u-nancy.fr) 2 Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Granada, 18071 Granada, Spain (e-mail: smontiel@goliat.ugr.es) Received: 2 February 2002; in final form: 1 August 2002 / Published online: 1 April 2003 – © Springer-Verlag 2003 Abstract. Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non- negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc. Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25 1. Introduction On a compact k-dimensional Riemannian spin manifold Q, real Killing spinors are characterized as eigenspinor fields of the Dirac operator associated with the smallest eigenvalues ±  k 4(k-1) R Q 0 , where R Q 0 denotes the infimum over Q of the scalar curvature. In other words, these are eigenspinor fields of the Dirac operator satisfying the limiting-case of the Friedrich inequality: λ 2 ≥ k 4(k - 1) R Q 0 . (1) Note that Inequality (1) is an immediate consequence of the spinorial Cauchy- Schwarz inequality and it is only interesting in the case where R Q 0 > 0. A The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when the boundary is at least 2-dimensional. Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001- 2967