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.

In this paper we study the asymptotic behavior of cylindrical ends in compact foliated 3-manifolds and give a sufficient condition for these ends to spiral onto a toral leaf.

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.

We begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.

In this note, we propose an extension of the compactness property for Kähler-Einstein metrics to critical metrics of Weyl functional on compact Kähler surfaces.

For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

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.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

We give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

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.

We give an account of the minimal volume of the plane, as defined by Gromov, and first computed by Bavard and Pansu. We also describe some related geometric inequalities.

We consider hypersurfaces of En+1 whose position vector x satisfies Δx = Ax + B, where Δ is the induced Laplacian, and prove that these are open parts of minimal hypersurfaces, hyperspheres or generalized circular cylinders.

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.

A complete classification is given of harmonic morphisms to a surface and conformal foliations by geodesics, with or without isolated singularities, of a simply-connected space form. The method is to associate to any such a holomorphic map from a Riemann surface into the space of geodesics of the space form. Properties such as nonintersecting fibres (or leaves) are translated into conditions on the holomorphic mapping which show it must have a simple form corresponding to a standard example.

Let 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ … denote the sequence of eigenvalues of the Laplacian of a compact minimal submanifold in a unit sphere. Yang and Yau obtained an upper bound on λn+1 in terms of λn and the sum λ1 + … + λn. In this note we shall prove an improved version of this upper bound by using the method of Hile and Protter.

In the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

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.

We study real hypersurfaces of a complex projection space and show that there are no such hypersurfaces with harmonic curvature on which the structure vector is principal.

We consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

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)≡ 〈, 〉).