We describe up to finite coverings causal flat affine complete Lorentzian manifolds such that the past and the future of any point are closed near this point. We say that these manifolds are strictly causal. In particular, we prove that their fundamental groups are virtually abelian. In dimension 4, there is only one, up to a scaling factor, strictly causal manifold which is not globally hyperbolic. For a generic point of this manifold, either the past or the future is not closed and contains a lightlike straight line.

Abstract. Several observations on spherical harmonics and their nodal sets are presented: a construction for harmonics with prescribed zeros; a natural representation for harmonics on S 2 ; upper and lower bounds for the nodal length and the inner radius (the upper bounds are sharp); the sharp upper bound for the number of common zeros of two spherical harmonics on S 2 ; the mean Hausdorff measure of the intersection of k nodal sets for harmonics of different degrees on S m , where k ≤ m (in particular, the mean number of common zeros of m harmonics).

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Let ∆u + λu = ∆v + λv = 0, where ∆ is the Laplace-Beltrami operator on a compact connected smooth manifold M and λ > 0. If H 1 (M ) = 0 then there exists p ∈ M such that u(p) = v(p) = 0. For homogeneous M , H 1 (M ) = 0 implies the existence of a pair u, v as above that has no common zero.