Trust-Region Methods on Riemannian Manifolds

A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.     

Subgradient Algorithm on Riemannian Manifolds

The subgradient method is generalized to the context of Riemannian manifolds. The motivation can be seen in non-Euclidean metrics that occur in interior-point methods. In that frame, the natural curves for local steps are the geodesies relative to the specific Riemannian manifold. In this paper, the influence of the sectional curvature of the manifold on the convergence of the method is discussed, as well as the proof of convergence if the sectional curvature is nonnegative.     

Proximal Calculus on Riemannian Manifolds

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M. We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M, as well as differentiability and geometrical properties of the distance function to a closed subset C of M. The first-named author was supported by a Marie Curie Intra-European Fellowship of the European Community, Human Resources and Mobility Programme under contract number MEIF CT2003-500927. The second-named author was supported by BFM2003-06420.     

Integrable Geodesic Flows on Riemannian Manifolds

We discuss the notion of geodesics and study the global behavior of geodesics on closed Riemannian manifolds. In particular, we emphasize the case of so-called integrable geodesic flows.     

Nonsmooth Optimization Techniques on Riemannian Manifolds

We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Riemannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a generalization of the Palais–Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds.     

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let : M N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kähler structure andM a Sasakian structure. (b) Let be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature . If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.     

Riemannian Submersions from Quaternionic Manifolds

In this paper we define the concept of quaternionic submersion, we study its fundamental properties and give an example.      

On the Calculus of Limiting Subjets on Riemannian Manifolds

In this paper fuzzy calculus rules for subjets of order two on finite dimensional Riemannian manifolds are obtained. Then a second order singular subjet derived from a sequence of efficient subsets of symmetric matrices is introduced. Employing fuzzy calculus rules for subjets of order two and various qualification assumptions based on a second order singular subjet, calculus rules for limiting subjets on a finite dimensional Riemannian manifold are obtianed.     

f-Harmonic Morphisms Between Riemannian Manifolds

f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970. In this paper, the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. The author also studies the f-harmonicity of conformal immersions.     

Cohomogeneity Two Actions on Flat Riemannian Manifolds

In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.     

Logarithmic Gradient of the Heat Kernel on Riemannian Manifolds

Mathematical Notes -     

An Incremental Subgradient Method on Riemannian Manifolds

In this paper, we propose and analyze an incremental subgradient method with a diminishing stepsize rule for a convex optimization problem on a Riemannian manifold, where the object function consisted of the sum of a large number of component functions. This type of function is useful in lots of fields. We establish an important inequality about the sequence generated by the method, provided the sectional curvature of the manifold is nonnegative. Using the inequality, we prove Proposition 3.1, and then we obtain some convergence results of the incremental subgradient method.     

Convexity of Domains of Riemannian Manifolds

In this paper the problem of the geodesic connectedness and convexity ofincomplete Riemannian manifolds is analyzed. To this aim, a detailedstudy of the notion of convexity for the associated Cauchy boundary iscarried out. In particular, under widely discussed hypotheses,we prove the convexity of open domains (whose boundaries may benondifferentiable) of a complete Riemannian manifold. Variationalmethods are mainly used. Examples and applications are provided,including a result for dynamical systems on the existence oftrajectories with fixed energy.     

An Estimate on Riemannian Manifolds of Dimension 4

We give an estimate of type sup × inf on Riemannian manifold of dimension 4 for a Yamabe type equation.     

Manifolds of Maps in Riemannian Foliations

Let (M,F) and (M,F) be two (compact or not) foliated manifolds, C _F (M, M) the space of smooth maps which send leaves into leaves. In this paper we prove that C _F (M, M) admits a structure of an infinite-dimensional manifold modeled on LF-spaces, provided that F is a Riemannian foliation or, more generally, when it admits an adapted local addition.     

On Saddle Submanifolds of Riemannian Manifolds

Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must intersect. Moreover, if M has positive k -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups _1(M) and ₁(M) is surjective.     

Hardy Spaces of Differential Forms on Riemannian Manifolds

Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H ^p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H ^p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ^∞ functional calculus and Hodge decomposition, are given.      

Stable Solutions of Elliptic Equations on Riemannian Manifolds

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows us to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.