Riemannian Manifold Research Articles

Constraint qualifications and optimality criteria for robust conic multiobjective semi-infinite programming problems with vanishing constraints on Riemannian manifolds

An Adaptive Cubic Regularization Inexact-Newton Method on Riemannian Manifolds

Abstract In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t , and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: $$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$ where h is the topological entropy of the geodesic flow. We also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.

Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds

Remarks on Hessian quotient equations on Riemannian manifolds

High-dimensional partially linear additive models on Riemannian manifolds

Smoothness estimation for Whittle–Matérn processes on closed Riemannian manifolds

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle S1, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras g(M), their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on SU(2), SU(2)/U(1), and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

We consider an area-minimizing integral current T of codimension higher than 1 in a smooth Riemannian manifold \Sigma . In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as the ‘singularity degree’. In this paper, we show that the set of points for which the singularity degree is strictly larger than 1 is (m-2) -rectifiable. In a subsequent work, we prove that the remaining flat singular points form a \mathcal{H}^{m-2} -null set, thus concluding that the singular set of T is (m-2) -rectifiable.

Abstract Choi-Wang obtained a lower bound of the first eigenvalue of the Laplacian on closed minimal hypersurfaces. On minimal hypersurfaces with boundary, Fraser-Li established an inequality giving a lower bound of the first Steklov eigenvalue as a counterpart of the Choi-Wang inequality. The Fraser-Li type inequality was obtained for manifolds with non-negative Ricci curvature. In this paper, we extend it to the setting of non-negative Ricci curvature with respect to the Wylie-Yeroshkin type affine connection. Our results apply to both weighted Riemannian manifolds with non-negative 1-weighted Ricci curvature and substatic triples.

Properties and examples of Kobayashi hyperbolic Riemannian manifolds

A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle TM. Recognizing that points in M are distributions and not just generic points in a manifold, TM can be extended to a Hilbert bundle HM. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization θ:M→Θ⊂Rk, θ^ is a distribution on Θ while its generalization gθ^=θ^−Eθ^ as an estimate is a function over Θ that indicates inconsistency between the model and data. As an estimator, gθ^ is a distribution of functions. Geometric properties of these functions describe statistical properties of gθ^. In particular, the expected slopes of gθ^ are used to define Λ(gθ^), the Λ-information of gθ^. The Fisher information I is an upper bound for the Λ-information: for all g, Λ(g)≤I. We demonstrate the utility of this geometric perspective using the two-sample problem.

Geodesics on Riemannian manifolds are precisely the locally length-minimizing curves, but their explicit description via simple functions is rarely possible. Geodesics of the simplest form, such as lines on Euclidean space and great circles on the round sphere, usually arise as orbits of one-parameter groups of isometries via Lie group actions. Manifolds where all geodesics are such orbits are called geodesic orbit manifolds (or g.o. manifolds), and their understanding and classification spans a quite long and continuous history in Riemannian geometry. In this paper, we classify the left-invariant g.o. metrics on the compact Lie group G 2 G_2 , using the nice representation theoretic behaviour of a class of Lie subgroups called (weakly) regular. We expect that the main tools and insights discussed here will facilitate further classifications of g.o. Lie groups, particularly of lower ranks.

Abstract This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace–Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as the Riemannian gradient flow of generalized harmonic energies and is thus governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher–Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof-of-concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.

We develop an innovative framework for financial modeling by integrating stochastic differential geometry with Ricci flow dynamics. In this model, asset prices evolve on a Riemannian manifold, and volatility is governed by a stochastic Ricci flow equation, producing a dynamically evolving volatility surface influenced by geometric curvature and stochastic forcing. We establish rigorous theoretical results on the existence and uniqueness of stochastic flows and demonstrate their impact on option pricing. Numerical simulations illustrate volatility clustering, geometric deformation, and realistic asset price behavior under curvature-driven uncertainty. This approach extends classical stochastic volatility models by capturing intrinsic geometric features of market dynamics, offering a robust tool for modeling turbulence, clustering, and complex financial phenomena with enhanced fidelity.

An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes df)\longrightarrow(M,g)$ where $f$ is a smooth function with gradient norm $<1$ on $(M,g)$. We construct new examples of identity harmonic maps. We define a symmetric tensor field on $M$ whose properties are related to the harmonicity of these identity maps.

In this paper, we investigate a class of metrics induced by $F$-natural metrics on the indicatrix bundle of a Finsler manifold. This class constitutes a four-parameter family that generalizes the well-known $g$-natural metrics on the unit tangent bundle of a Riemannian manifold. Within this framework, we construct a three-parameter family of contact metric structures whose associated metrics are $F$-natural, and we establish that, in contrast to the Riemannian case —where all such $K$-contact metrics on unit tangent bundles are necessarily Sasakian— the corresponding structures in the Finslerian setting can be $K$-contact without being Sasakian. Furthermore, we provide a characterization of Finsler manifolds with positive constant flag curvature via the existence of $K$-contact structures on their indicatrix bundles.

On cutoff via rigidity for high dimensional curved diffusions

We consider solutions to some semilinear elliptic equations on complete noncompact Riemannian manifolds and study their classification as well as the effect of their presence on the underlying manifold. When the Ricci curvature is nonnegative, we prove both the classification of positive solutions to the critical equation and the rigidity for the ambient manifold. The same results are obtained when we consider solutions to the Liouville equation on Riemannian surfaces. The results are obtained via a suitable P -function whose constancy implies the classification of both the solutions and the underlying manifold. The analysis carried out on the P -function also makes it possible to classify nonnegative solutions for subcritical equations on manifolds enjoying a Sobolev inequality and satisfying an integrability condition on the negative part of the Ricci curvature. Some of our results are new even in the Euclidean case.

Higher dimensional Q-curvature problems on Riemannian manifolds