If an (n + 2)-dimensional Lorentzian manifold is indecomposable, but non-irreducible, then its holonomy algebra is contained in the parabolic algebra (R⊕so(n)).Rn. We show that its projection onto so(n) is the holonomy algebra of a Riemannian manifold. This leads to a classification of Lorentzian holonomy groups and implies that the holonomy group of an indecomposable Lorentzian spin manifold with parallel spinor equals to G . Rn where G is a product of SU(p), Sp(q), G2 or Spin(7).

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Let<i>M</i> be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass of<i>M</i> as viewed from spatial infinity (the ADM mass) must be positive unless<i>M</i> is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

We prove several formulas related to Hodge theory and theKodaira– Spencer–Kuranishi deformation theory of Kähler manifolds. As applications, wepresent a construction of globally convergent power series of integrableBeltrami differentials on Calabi–Yau manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of Calabi–Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kähler manifolds.