Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

We introduce complex differential geometry twisted by a real line bundle. This provides a new approach to understand the various real objects that are associated with an anti-holomorphic involution. We also generalize a result of Greenleaf about real analytic sheaves from dimension 2 to higher dimensions.

In this paper we prove that minimal 3-spheres of CR type with constant sectional curvature c in the complex projective space CPn are all equivariant and therefore the immersion is rigid. The curvature c of the sphere should be c = 1/(m2-1) for some integer m≥ 2, and the full dimension is n = 2m2-3. An explicit analytic expression for such an immersion is given.

In a convex domain K in ℝd , a transmitter and a receiver are placed at random according to the uniform distribution. The statistics of the power received by the receiver is an important quantity for the design of wireless communication systems. Bounds for the moments of the received power are given, which depend only on the volume and the surface area of the convex domain.

We apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its Ln norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from -∞ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.

We study the harmonicity of maps to or from cosymplectic manifolds by relating them to maps to or from Kähler spaces.

In this paper we establish an integral formula for compact hypersurfaces in non-flat space forms, and apply it to derive some interesting applications. In particular, we obtain a characterization of geodesic spheres in terms of a relationship between the scalar curvature of the hypersurface and the size of its Gauss map image. We also derive an inequality involving the average scalar curvature of the hypersurface and the radius of a geodesic ball in the ambient space containing the hypersurface, characterizing the geodesic spheres as those for which equality holds.

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd , the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.

We determine the naturally reductive homogeneous real hypersurfaces in the family of curvature-adapted real hypersurfaces in quaternionic projective space HPn(n ≥ 3). We conclude that the naturally reductive curvature-adapted real hypersurfaces in HPn are Q-quasiumbilical and vice-versa. Further, we study the same problem in quaternionic hyperbolic space HHn(n ≥ 3).

Let be the mean normal measure of a stationary random set Z in the extended convex ring in ℝd . For k ∈ {1,…,d-1}, connections are shown between and the mean of . Here, the mean is understood to be with respect to the random isotropic k-dimensional linear subspace ξk and the mean normal measure of the intersection is computed in ξk . This mean to be well defined, a suitable spherical lifting must be applied to before averaging. A large class of liftings and their resulting means are discussed. In particular, a geometrically motivated lifting is presented, for which the mean of liftings of determines uniquely for any fixed k ∈ {2,…,d-1}.

Recently, systematic sampling on the circle and the sphere has been studied by Gual-Arnau and Cruz-Orive (2000) from a design-based point of view. In this note, it is shown that their mathematical model for the covariogram is, in a model-based statistical setting, a special case of the p-order shape model suggested by Hobolth, Pedersen and Jensen (2000) and Hobolth, Kent and Dryden (2002) for planar objects without landmarks. Benefits of this observation include an alternative variance estimator, applicable in the original problem of systematic sampling. In a wider perspective, the paper contributes to the discussion concerning design-based versus model-based stereology.

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

Recently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms.

In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.

The set covariance of a dead leaves model, constructed from hard spheres of constant diameter, is calculated analytically. The calculation is based on the covariance of a single sphere and on the pair correlation function of the centres of the spheres. There exist applications in the field of random sequential adsorption and in the interpretation of small-angle scattering experiments.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

Let Mn, n ≥ 3, be a complete oriented minimal hypersurface in Euclidean space Rn+1. It is shown that, if the total scalar curvature on M is less than the n/2 power of 1/2Cs, where Cs is the Sobolev constant for M, and the square norm of the second fundamental form is a L2 function, then M is a hyperplane.

In this paper we construct a family of variational families for a Legendrian embedding, into the 1-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embdeddings, by discretising the action functional. We compute the second variation of a generating funciton obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generlisation of the Morse inequalities thus refining a theorem of Chekanov.

In this paper we prove a uniqueness theorem for minimal discs in R3 spanning a polygonal boundary.

Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in ℝd . Recently, the theory of systematic sampling on ℝ has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (𝕊1) and the semicircle (ℍ1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (𝕊2). In this paper we adapt the theory available for non-periodic functions of bounded support on ℝ to periodic functions, and thereby to 𝕊1 and ℍ1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in 𝕊2. The paper starts with a historical perspective, and ends with suggestions for further research.