Search Results (38)

We consider the reciprocal cost function $J\left(x\right)={\displaystyle \frac{1}{2}}(x+x^{-1})-1$ and its n-dimensional extension $J(x_1,\dots ,x_n)={\displaystyle \frac{1}{2}}(R+R^{-1})-1,R={\displaystyle \prod _{i=1}^n}{x}_i^{{\alpha }_i},\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{R}}^n\setminus \left\{0\right\}.$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of J has a rank of one at every point. The associated Hessian geometry is degenerate and does not define a Riemannian metric. To obtain a nondegenerate geometric structure, we introduce a family of Hessian metrics $h_{\lambda }$. Combining the rank-one tensor with the Hessian metric $h_{\lambda }$, we construct a $(1,1)$-tensor field $A_{\lambda }$. Its trace normalization defines a projector $P_{\lambda }$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_{\lambda }$ and the induced structures, including eigendistributions, parallelism, integrability, and curvature. The construction is given in an arbitrary dimension, and explicit formulas are obtained in the two-dimensional case. In particular, we show that the projector $P_{\lambda }$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection ${\nabla }^{\lambda }$ of the Hessian metric $h_{\lambda }$. Full article

In this work, we study the relationships between affine manifolds and complex manifolds. We prove that a linear connection and a Riemannian metric on an affine manifold M of dimension n induce a complex structure and a Hermitian metric on both the product $M\times {\mathbb{R}}^n$ and the tangent bundle $TM$. We also discuss some geometric relations among the affine manifold M and the Hermitian manifolds $M\times {\mathbb{R}}^n$ and $TM$. As an application in analysis, we obtain the generalized maximum principle on complete affine Riemannian manifolds, which can be used to study partial differential equations. It is worth noting that Hessian manifolds, as a special case of affine manifolds, have broad application potential in statistics and information geometry. Full article

This article explores properties of conformal vector fields on a Riemannian manifold, focusing on conditions that lead to the manifold being isometric to the Euclidean space. Given a conformal vector $\zeta$ with conformal factor $\sigma$ on a Riemannian manifold $\left(N,g\right)$, there is naturally associated a skew-symmetric tensor $\chi$ to $\zeta$ called the essential tensor of $\zeta$. It is shown that the essential tensor $\chi$ plays a vital role in our study. We intend to analyze when a conformal vector field becomes a Killing vector field. In a first result of this article, we obtain a necessary and sufficient geometric condition on a complete and connected Riemannian manifold $\left(N,g\right)$ admitting a conformal vector field $\zeta$ so that $\zeta$ is a Killing vector field. In the rest of the article, we obtain characterizations of a Euclidean space using conformal vector fields. In the first such result, it is shown that an n-dimensional complete and connected Riemannian manifold $\left(N,g\right)$, $n>2$ admits a conformal vector field $\zeta$ with conformal factor $\sigma \ne 0$ and essential tensor $\chi$ such that the affinity tensor of $\nabla \sigma$ is zero, the function $\zeta \left(\sigma \right)$ is a constant, and $\zeta$ annihilates $\chi$ if and only if $\left(N,g\right)$ is isometric to the Euclidean space $E^n$. Similarly, in a second characterization of the Euclidean space $E^n$ using a conformal vector field $\zeta$, we use the following conditions: $\zeta$ annihilates the Ricci operator S, $\nabla \sigma$ annihilates $\chi$, and the vector field $\chi \left(\zeta \right)$ is incompressible. Finally, we consider a conformal vector field $\zeta$ with conformal factor $\sigma \ne 0$ and essential tensor $\chi$ on a complete and connected Riemannian manifold $\left(N,g\right)$ such that the Hessian operator $H_{\sigma }$ is invariant under the local flow of $\zeta$ so that the function $\zeta \left(\sigma \right)-{\sigma }^2$ is a subharmonic function and $\zeta$ annihilates $\chi$, and show that $\left(N,g\right)$ is isometric to the Euclidean space $E^n$. The converse holds as well. Full article

This work formulates semiclassical gravity within a causal-diamond framework where a finite-resolution boundary provides the edge structure for a local Wheeler–DeWitt description. Because the diffeomorphism-invariant Hilbert space does not factorize, each diamond is equipped with a boundary-completed algebra ${\mathfrak{A}}_O$, ensuring the operational state ${\rho }_O$ and the semiclassical reference family ${\sigma }_O[\Lambda ]$ share identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional $S_{\mathrm{rel}}({\rho }_O\parallel {\sigma }_O[\Lambda ])$ quantifies the mismatch between data and reference. This yields the minimal operational axioms defining subsystems, intrinsic clocks, and regulated observables in a finite-resolution, background-independent setting. The topology-locked boundary capacity budget fixes an effective channel multiplicity $N\approx 1.23\times {10}^{11}$. Calibrating its coherent fraction to Newton’s constant determines a matching scale $M_s\approx 3.02\times {10}^{13}\mathrm{GeV}$. In the modular/KMS regime, the relative-entropy Hessian (Kubo–Mori metric) block-diagonalizes into orthogonal tensor, vector, and scalar response sectors. A heat-kernel expansion on the fixed $S^3\times S^1$ history manifold maps this near-equilibrium response to a matching-scale effective field theory, yielding the Einstein–Hilbert tensor structure, Yang–Mills susceptibilities, and leading mass deformations. Vector and scalar responses remain intensive, while the tensor response scales extensively with coherent channel multiplicity. The fixed modular protocol and quantized boundary currents imply ${\alpha }^{-1}\left(M_s\right)=4\pi k$ at integer levels k, while the reduced $R^2$ plateau sector yields linked cosmological targets: $n_s\simeq 0.965$, $r\simeq 0.0038$, and $A_s\simeq 2.1\times {10}^{-9}$. Translations between causal diamonds act as completely positive trace-preserving (CPTP) updates. The resulting open-modular Walsh filtration selects the three-dimensional degree-one sector as the algebraic basis for family structure. Treating continuum fields as the structured response of a finite boundary, the framework yields correlated, falsifiable relations for gravitational stiffness, gauge response, plateau cosmology, and threefold matter-sector organization from one minimal operational architecture. Full article

We find restrictions on a trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ so that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. In that, first we show that if the vector $\mathbf{u}$ of the trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is an affine conformal vector with affine potential $\alpha \ne 0$ and the condition $\mathbf{u}\left(\alpha \right)=-{\beta }^2$ holds, necessarily implies $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. Similarly, it is shown that if the vector $\mathbf{u}$ of the trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ is a projective vector and the sectional curvatures of the plane sections containing $\mathbf{u}$ are positive constant, then $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. Finally, we find certain generic conditions on a 3-dimensional Riemannian manifold $\left(M^3,g\right)$ possessing a trans-Sasakian structure $\left(F,\mathbf{u},\gamma ,\alpha ,\beta \right)$ so that $\left(M^3,g\right)$ is homothetic to a Sasakian manifold. Full article

In dually flat manifolds, there is a deep connection between gradient flows and pregeodesics. This was one of the many important contributions of Amari to information geometry. In this paper, we extend the study of this relationship to general Riemannian manifolds. Our result does not impose conditions of flatness on the connection or symmetry on its non-metricity tensor, thus broadening the geometric setting beyond Hessian manifolds. Within this framework, we provide a criterion for comparing relaxation along two different gradient descent curves of a function, formulated in terms of the non-metricity tensor of a connection for which the gradient curves are pregeodesics. We use it to study Gaussian chains, whose relaxation trajectories coincide with gradient descent curves in the space of Gaussian distributions. Thus, we recover a recent result that establishes a universal asymmetry: warming up is faster than cooling down. Our work illustrates how geometric insights rooted in Amari’s legacy offer new perspectives for optimization problems and stochastic processes. Full article

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are gauge structures on these vector bundles. We are interested in the dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e., properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations, three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection an affine connection? (P2-Riemannian Geometry): When is a Koszul connection a metric connection? (P3-Fedosov Geometry): When is a Koszul connection a symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of how to produce labeled foliations the most studied of which are Riemannian foliations. On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemented to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce Koszul Homological Series. This notion is a machine for converting obstructions whose nature is vector space into obstructions whose nature is homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2), and (P3). In the abundant literature on Riemannian foliations, we have only cited references directly related to the open problems which are studied using the tools which are introduced in this work. Thus, the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How does one produce Riemannian foliations? See our Theorems 12 and 13, which are fruits of a happy conjunction between gauge geometry and differential topology. Full article

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the statistical manifold on the Orlicz space $L_0^{\mathsf{\Phi }}\left(P_f\right)$ (the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space ($T_{f}M=S\oplus S^{\perp }$), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement ($S^{\perp }$). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as $G_f$, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem ($H_G\left(f\right)=\mathrm{Tr}\left(G_f\right)$) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant $(0,2)$-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection). Full article

Recently, the adaptive regularized proximal quasi-Newton (ARPQN) method has demonstrated a strong performance in solving composite optimization problems over the Stiefel manifold. However, its reliance on first-order information limits its applicability to scenarios where gradient and Hessian evaluations are unavailable or costly. In this paper, we propose a zeroth-order adaptive regularized proximal quasi-Newton method (ZO-ARPQN) for black-box composite optimization over Riemannian manifolds, particularly the Stiefel and symmetric positive definite (SPD) manifolds. The proposed method estimates the Riemannian gradient and curvature information through randomized one-point finite-difference approximations and adaptively updates a regularized quasi-Newton matrix to capture the local manifold geometry. Theoretically, we established global convergence and complex analyses under mild assumptions. More importantly, by incorporating curvature-aware regularization and random perturbations in the proximal quasi-Newton framework, we proved that ZO-ARPQN can escape strict saddle points with a high probability. This guarantees convergence to a stationary point, even in the absence of explicit gradients. Extensive numerical experiments were conducted on manifold-constrained problems, including sparse PCA and robot stiffness tuning. These demonstrated that ZO-ARPQN shows a competitive convergence behavior compared with other state-of-the-art Riemannian optimization methods, while requiring only function evaluations. Full article

Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661–674] characterized invertible reverse-ordering transforms in the space of lower, semi-continuous, extended, real-valued convex functions as affine deformations of the ordinary Legendre transform. In this work, we first prove that all those generalized Legendre transforms of functions correspond to the ordinary Legendre transform of dually corresponding affine-deformed functions. In short, generalized convex conjugates are ordinary convex conjugates of dually affine-deformed functions. Second, we explain how these generalized Legendre transforms can be derived from the dual Hessian structures of information geometry. Full article

This paper studies $(p,q)$-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the $(p,q)$-energy functional. Second, we analyze weakly conformal and horizontally conformal $(p,q)$-harmonic maps and prove Liouville results for $(p,q)$-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the $(p,q)$-$SSU$ manifold and prove that non-constant stable $(p,q)$-harmonic maps do not exist. Full article

This editorial presents 24 research articles published in the Special Issue entitled Geometry of Manifolds and Applications of the MDPI Mathematics journal, which covers a wide range of topics from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in many branches of theoretical and applied mathematical studies, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as complex space forms, metallic Riemannian space forms, Hessian manifolds of constant Hessian curvature; optimal inequalities for submanifolds, such as generalized Wintgen inequality, inequalities involving $\delta$-invariants; homogeneous spaces and Poisson–Lie groups; the geometry of biharmonic maps; solitons (Ricci solitons, Yamabe solitons, Einstein solitons) in different geometries such as contact and paracontact geometry, complex and metallic Riemannian geometry, statistical and Weyl geometry; perfect fluid spacetimes [...] Full article

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function $\psi$ and the vector field $\zeta$ dual to $d\psi$ does not change its causal character, then P is homothetic, or $\zeta$ is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field. Full article

In this study, we investigate the tangent bundle $TM$ of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection $\tilde{\nabla }$. Our primary goal is to establish the necessary and sufficient conditions for $TM$ to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for $TM$ to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to $TM$, alongside specific constraints on the conformal factor $\lambda$ and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve $\lambda$ and the Hessian of the potential function. Similarly, for $TM$ to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of $\lambda$, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article

The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry of the manifolds under consideration. In this article, we use the gauge equation to introduce spectral sequences that are further specialized to Hessian structures. Full article