We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn in K, i.e. the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the d > 1 case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

In this paper we determine the metric dimension of $n$-dimensional metric $(X,G)$-manifolds. This category includes all Euclidean, hyperbolic and spherical manifolds as special cases.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

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.

Given an open set $\Omega$ of compact closure in $\mathbb{R}^m$, the classical Dirichlet problem is to extend a given continuous function $\psi : \partial \Omega \to \mathbb{R}$ to a continuous function $\phi : \overline \Omega \to \mathbb{R}$ such that $\phi$ is harmonic (that is, satisfies the Laplace equation) in $\Omega$. The set $\Omega$ is termed regular if the Dirichlet problem has a (necessarily unique) solution for any continuous boundary function $\psi$. For example, every simply connected planar domain is regular (but may have a 'bad' boundary $\partial\Omega$, for instance, a fractal).

In this article it is shown that, when $\Omega$ is regular (in the above sense), every continuous map $\psi$ from $\partial\Omega$ to a simply connected complete Riemannian manifold $(N, h)$ of sectional curvature at most 0 has a unique continuous extension $\phi : \overline\Omega \to N$ which is harmonic in $\Omega$. This is done with $\mathbb{R}^m$ replaced more generally by an $m$-dimensional Riemannian manifold $(M, g)$.

The proof relies on the unique solvability of the corresponding variational Dirichlet problem (for any open set $\Omega \Subset M$). And for that, the above target manifold $N$ can be replaced more generally by any simply connected complete geodesic space $Y$ of curvature at most 0 in the sense of A. D. Alexandrov. Assuming that $M$ satisfies the Poincaré inequality, we show that, for any map $\psi : M \to Y$ of finite energy in the sense of N. J. Korevaar and R. M. Schoen, there exists a unique map $\phi : M \to Y$ with $\phi = \psi$ on $M \setminus \Omega$ such that $\phi$ minimizes the energy of all maps $M \to Y$ which agree with $\psi$ on $M \setminus \Omega$. If $\Omega$ is regular then $\phi$ is continuous at any point of $\partial\Omega$ at which $\psi$ is continuous. For a Lipschitz (and hence regular) domain $\Omega \Subset M$, existence and uniqueness of the variational solution $\phi$ was obtained by Korevaar and Schoen, and earlier for suitable polyhedral targets $Y$ by Gromov and Schoen.

Instead of the Riemannian manifold $M$ our domain space can still more generally be an admissible Riemannian polyhedron (as studied in the recent Cambridge Tract by J. Eells and the present author); the variational solution $\phi$ is then in general only Hölder continuous in $\Omega$.

The proofs of the stated results of this article rely in part on potential theory relative to the fine topology of H. Cartan.

Some curvature conditions about the geodesics emanating from a submanifold are obtained. These conditions are used to to study the topological and geometric properties of the ambient spaces which admit some minimal submanifolds.