We consider Riemannian orbifolds with Ricci curvature nonnegative outside a compact set and prove that the number of ends is finite. We also show that if that compact set is small then the Riemannian orbifolds have only two ends. A version of splitting theorem for orbifolds also follows as an easy consequence.

Let M be a compact flat Riemannian manifold of dimension n, and Γ its fundamental group. Then we have the following exact sequence (see [1])

where Zn is a maximal abelian subgroup of Γ and G is a finite group isomorphic to the holonomy group of M. We shall call Γ a Bieberbach group. Let T be a flat torus, and let Ggr act via isometries on T; then ┌ acts isometrically on × T where is the universal covering of M and yields a flat Riemannian structure on ( × T)/Γ. A flat-toral extension (see [9, p. 371]) of the Riemannian manifold M is any Riemannian manifold isometric to ( × T)/Γ where T is a flat torus on which Γ acts via isometries. It is convenient to adopt the convention that a single point is a 0-dimensional flat torus. If this is done, M is itself among the flat toral extensions of M. Roughly speaking, this is a way of putting together a compact flat manifold and a flat torus to make a new flat manifold the dimension of which is the sum of the dimensions of its constituents. It is, more precisely, a fibre bundle over the flat manifold with a flat torus as fibre.

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points). Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. The purpose of this paper is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling. Relevant mechanisms to increase sample size, compatible with stereological practice, are considered.

Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.

All manifolds in this paper are assumed to be closed, oriented and smooth.

A contact structure on a (2n + l)-dimensional manifold M is a maximally non-integrable hyperplane distribution D in the tangent bundle TM, i.e., D is locally denned as the kernel of a 1-form α satisfying α ۸ (da)n ۸ 0. A global form satisfying this condition is called a contact form. In the situations we are dealing with, every contact structure will be given by a contact form (see [5]). A manifold admitting a contact structure is called a contact manifold.

Simply connected conformally flat Riemannian manifolds are characterized as hypersurfaces in the light cone of a standard flat Lorentzian space, transversal to its generators. Some applications of this fact are given.

Let M be a smooth surface in Euclidean space E3 and L the Weingarten map. The fundamental forms I1, I2, I3,… on M are defined in terms of L and the usual inner product 〈, 〉 of E3 as follows. If X and Y are in the tangent space TPM of M (Pε M), then I1(X, Y) = 〈X, Y), I2(X, Y) = 〈LX, Y〉, I3(X, Y) = 〈L2X, Y), etc. Moreover, if M is convex, i.e., the Gaussian curvature K = k1k2, where ki, (i = l,2) are the principal curvatures of M, is everywhere positive, then one can also define on M the forms I0(X, Y) = 〈L−1X, Y), I−1,(X, Y) = 〈L−2X, Y), I−2(X, Y) = 〈 L−3X, Y) etc., where L−1 is the inverse of L. Since L is self-adjoint, the forms Im are, for any integer m, symmetric bilinear functions on TPM × TPM. Furthermore Im are C∞ in the sense that if X and Y are vector fields with domain A ⊂ M, then 〈 LmX, Y〉P = 〉LmXP, YP) is a C∞ real function on A. If the convex surface M is appropriately oriented, then the forms Im define metrics on M, which we also denote by 〈, 〉m (〈, 〉1)≡ 〈, 〉).

Let Mn be a smooth, compact and strictly convex, embedded hypersurface of Rn + 1 (n ≥ 1), an ovaloid for short. By “strictly convex” we mean that the Gauss-Kronecker curvature where ki are the principal curvatures with respect to the inner unit normal field, is everywhere positive. It is well knpwn [5, p. 41] that, for such a hypersurface, the spherical-image mapping is a diffeomorphism onto the unit hypersphere. Furthermore, Mn is the boundary of an open bounded convex body, which we shall call the interior of Mn.

Consider two convex bodies K, K′ in Euclidean space En and paint subsets β, β′ on the boundaries of K and K′. Now assume that K′ undergoes random motion in such a way that it touches K.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.

1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

see [4; p. 214]. Blaschke's formula [4; p. 243] gives

as the measure, to within a normalization, of overlapping positions of K1 relative to K0.