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Abstract. The restriction of a parallel spinor on some spin manifold Z to a hypersurface M ⊂ Z is a generalized Killing spinor on M . We show, conversely, that in the real analytic category, every spin manifold (M, g) carrying a generalized Killing spinor ψ can be isometrically embedded as a hypersurface in a spin manifold carrying a parallel spinor whose restriction to M is ψ. We also answer negatively the corresponding question in the smooth category.

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is purely discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.

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We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z o Γ,Σ (λ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ\H 2n+1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η(D) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ\H 2n+1 and that η(D) = 1 πi log Z o Γ,Σ (0), thus extending Millson's formula to this setting. We also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil-Petersson metric on Schottky space under some assumption on the exponent of convergence of Poincaré series for the group Γ.