Riemannian Setting Research Articles

A fundamental differential system of Riemannian geometry

We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree n associated to any given oriented Riemannian manifold M of dimension n + 1 . The framework is that of the tangent sphere bundle of M . We generalise to a Riemannian setting some results from the theory of hypersurfaces in flat Euclidean space. We give new applications and examples of the associated Euler–Lagrange differential systems.

Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs.

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min–max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min–max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.

Approximation of Densities on Riemannian Manifolds.

Finding an approximate probability distribution best representing a sample on a measure space is one of the most basic operations in statistics. Many procedures were designed for that purpose when the underlying space is a finite dimensional Euclidean space. In applications, however, such a simple setting may not be adapted and one has to consider data living on a Riemannian manifold. The lack of unique generalizations of the classical distributions, along with theoretical and numerical obstructions require several options to be considered. The present work surveys some possible extensions of well known families of densities to the Riemannian setting, both for parametric and non-parametric estimation.

A geometric perspective on the Piola identity in Riemannian settings

The Piola identity \begin{document}$ \operatorname{div}\; \operatorname{cof} \;\nabla f = 0 $\end{document} is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.

A nonlinear elliptic eigenvalue–transmission problem with Neumann boundary condition

Let $$\varOmega \subset \mathbb {R}^N$$ , $$N\ge 2$$ , be a bounded domain which is divided into two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ . Consider in $$\varOmega $$ an eigenvalue–transmission problem associated with the p-Laplacian acting in $$\varOmega _1$$ and the q-Laplacian acting in $$\varOmega _2$$ , $$1<p<q$$ , with Dirichlet–Neumann conditions on the interface separating the two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ [see (1.1)]. The main result Theorem 2.1 states the existence of a sequence of eigenvalues for this eigenvalue problem. The proof is based on the Lusternik–Schnirelmann principle. Using the method of Lagrange multipliers for constrained minimization problems, we show (see Theorem 2.2) that if $$2\le p<q$$ then there exists an eigenfunction in any set of the form $$\begin{aligned} \left\{ u\in W^{1,p}(\varOmega );\, u|_{\varOmega _2}\in W^{1,q}(\varOmega _2), \, \frac{1}{p}\int _{\varOmega _1}|u|^p + \frac{1}{q}\int _{\varOmega _2}|u|^q = \alpha \right\} , \quad \alpha >0. \end{aligned}$$ The case of Robin conditions on $$\partial \varOmega $$ and the Riemannian setting are also addressed.

A two-phase-like proximal point algorithm in domains of positivity

This paper improves a decomposition-like proximal point algorithm, developed for computing minima of nonsmooth convex functions within a framework of symmetric positive semidefinite matrices, and extends it to domains of positivity of reducible type, in a nonlinear sense and in a Riemannian setting. Several computational experiments with weighted Lp (p=1,2) centers of mass are performed to demonstrate the practical feasibility of the method.

Quantum confinement on non-complete Riemannian manifolds

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure \omega , possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V\in L^2_\mathrm {loc}(M) . The main merit of this paper is the identification of an intrinsic quantity, the effective potential V_\mathrm {eff} , which allows to formulate simple criteria for quantum confinement. Let \delta be the distance from the possibly non-compact metric boundary of M . A simplified version of the main result guarantees quantum completeness if V\ge -c\delta^2 far from the metric boundary and V_\mathrm {eff}+V\ge \frac3{4\delta^2}-\frac{\kappa}{\delta}, \quad \mathrm {close \:to \:the \:metric \:boundary}. These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of M ; (ii) generalize the Kalf–Walter–Schmincke–Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace–Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace–Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [9].

The Degenerate Special Lagrangian Equation on Riemannian Manifolds

Abstract We show that the degenerate special Lagrangian equation (DSL), recently introduced by Rubinstein–Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi–Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey–Lawson, in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence continuous geodesics in the space of positive Lagrangians.

Riemannian and Lorentzian flow-cut theorems

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut (MFMC) theorem for boundary regions, applied recently to develop a ‘bit-thread’ interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous MFMC theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth’s theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.

Almost-global tracking for a rigid body with internal rotors

Inspired by recent work on a PID-like tracking control law for an interconnected mechanical system on a Lie group, this article presents a novel tracking control law for such mechanical systems, adapted from the one proposed by the authors for systems evolving on a Riemannian manifold setting. The interconnected mechanical system consists of a rigid body with three internal rotors for attitude actuation. The novelty of the control law lies in viewing the rigid body as a simple mechanical system (SMS) by incorporating the additional quadratic velocity terms arising from the interconnection into the control law, and then proceeding with a straightforward stabilizing PID tracking synthesis. The novel algorithm achieves tracking of feasible trajectories from almost all initial conditions.

A New Class of First Order Displacement Convex Functionals

New displacement convex functionals on $S^1$ are presented, and a novel Eulerian extension of the Otto calculus is developed. The proof of the displacement convexity of these functionals relies on this Eulerian calculus, which extends the Otto calculus from a purely Riemannian setting to general Lagrangians and provides a systematic way to compute displacement Hessians of functionals involving derivatives of densities. This resolves, in the special cases of $\mathbb{R}$ and $S^1$, a problem posed by Villani in Open Problem 15.11 of [Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009].

Space-times paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Holder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.

Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

Abstract Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} , which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} ) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.

A Note on the Almansi Property

The first goal of this note is to study the Almansi property on an m-dimensional model in the sense of Greene and Wu and, more generally, in a Riemannian geometric setting. In particular, we shall prove that the only model on which the Almansi property is verified is the Euclidean space $${\mathbb R}^m$$ . In the second part of the paper we shall study Almansi’s property and biharmonicity for functions which depend on the distance from a given submanifold. Finally, in the last section we provide an extension to the semi-Euclidean case $${\mathbb R}^{p,q}$$ which includes the proof of the classical Almansi property in $${\mathbb R}^m$$ as a special instance.

Finite length spectra of random surfaces and their dependence on genus

The main goal of this paper is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we study the probability distribution of the systole of random surfaces in a hyperbolic and a more general Riemannian setting. In the hyperbolic setting we are able to determine the limit of the probability distribution for the number of triangles tending to infinity and in the Riemannian setting we derive bounds.

Last Multipliers for Riemannian Geometries, Dirichlet Forms and Markov Diffusion Semigroups

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carre du champ associated to the infinitesimal generator of the semigroup.

A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds

In this paper, an optimization method for nonsmooth locally Lipschitz functions on complete Riemannian manifolds is presented. The method is based on approximating the subdifferential of the cost function at every iteration by the convex hull of transported gradients from tangent spaces at randomly generated nearby points to the tangent space of the current iterate and can hence be seen as a generalization of the well known gradient sampling algorithm to a Riemannian setting. A convergence result is obtained under the assumption that the cost function is bounded below and continuously differentiable on an open set of full measure and that the employed vector transport and retraction satisfy certain conditions, which hold, for instance, for the exponential map and parallel transport. Then with probability one the algorithm produces iterates at which the cost function is differentiable, and each cluster point of the iterates is a Clarke stationary point. Modifications yielding only $\varepsilon$-stationary ...

Random Pairwise Gossip on $\text{CAT}(\kappa)$ Metric Spaces

In the context of sensor networks, gossip algorithms are a popular, well established technique for achieving consensus when sensor data are encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, exhibits a very simple metric property that does not rely on any differential structure. This property strongly suggests that gossip algorithms could be studied on a broader family than Riemannian manifolds. And it turns out that, indeed, (local) convergence is guaranteed as soon as the data space is a mere $\text{CAT}(\kappa)$ metric space. We also study convergence speed in this setting and establish linear rates for $\text{CAT}(0)$ spaces, and local linear rates for $\text{CAT}(\kappa)$ spaces with $\kappa>0$ . Numerical simulations on several scenarii, with corresponding state spaces that are either Riemannian manifolds—as in the problem of positive definite matrices consensus—or bare metric spaces—as in the problem of arms consensus—validate the results. This shows that our metric approach not only allows for a simpler and more general mathematical analysis but also paves the way for new kinds of applications that go beyond the Riemannian setting.