spaces) are defined as those homogeneous Riemannian spaces ( M = G ∕ H , g ) whose geodesics are orbits of one-parameter subgroups of G.

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G.

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Also, we consider special almost geodesic mappings of the second type between Eisenhart’s generalized Riemannian spaces as well as between generalized classical (elliptic) and hyperbolic Kähler spaces.

Special Almost Geodesic Mappings of the Second Type Between Generalized Riemannian Spaces

We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces.

On Conformal and Concircular Diffeomorphisms of Eisenhart’s Generalized Riemannian Spaces

By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.

Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices

Empirical investigations on synthetic data and real-world motor imagery EEG data demonstrate that the performance of the proposed generalized learning Riemannian space quantization can significantly outperform the Euclidean GLVQ, generalized relevance LVQ (GRLVQ), and generalized matrix LVQ (GMLVQ).

Generalized Learning Riemannian Space Quantization: A Case Study on Riemannian Manifold of SPD Matrices

We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting.

On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

In the case when such hypersurface is a surface with constant mean curvature in a semi-Riemannian space form, we prove that it has an intrinsic Killing vector field.

Semi Riemannian hypersurfaces with a canonical principal direction

It is shown that in the case of a Riemannian space Vn, in which the group Gr acts simply transitively, the algebra of symmetry operators of the n-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group Gr.

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn.

Infinitesimal conformal transformations in the Riemannian space of the second approximation for a space of non-zero constant curvature

By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.

Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices

Empirical investigations on synthetic data and real-world motor imagery EEG data demonstrate that the performance of the proposed generalized learning Riemannian space quantization can significantly outperform the Euclidean GLVQ, generalized relevance LVQ (GRLVQ), and generalized matrix LVQ (GMLVQ).

Generalized Learning Riemannian Space Quantization: A Case Study on Riemannian Manifold of SPD Matrices

10.1109/ACCESS.2020.3048683

The aligned training data from chosen source subjects are used for creating a classification model based on either spatial covariance matrices in Riemannian space or common spatial pattern algorithm in Euclidean space.

Transfer Learning Based on Hybrid Riemannian and Euclidean Space Data Alignment and Subject Selection in Brain-Computer Interfaces

“Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

Quantity in Quantum Mechanics and the Quantity of Quantum Information

10.3103/S0025654421030134

— The article presents a new interpretation of the basic geometric concept of general theory of relativity, according to which gravity is associated not with the curvature of the Riemannian space generated by it, but with deformations of this space.

ANALOGY BETWEEN THE EQUATIONS OF ELASTICITY AND THE GENERAL THEORY OF RELATIVITY

We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting.

On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

10.1007/S40590-021-00361-Z

In the case when such hypersurface is a surface with constant mean curvature in a semi-Riemannian space form, we prove that it has an intrinsic Killing vector field.

Semi Riemannian hypersurfaces with a canonical principal direction

10.1016/j.neunet.2021.04.024

By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.

Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices

10.4310/CAG.2021.V29.N1.A3

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry.

Model spaces in sub-Riemannian geometry

10.1007/s00022-021-00574-7

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve.

Curves in three dimensional Riemannian space forms

It is shown that in the case of a Riemannian space Vn, in which the group Gr acts simply transitively, the algebra of symmetry operators of the n-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group Gr.

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

, 2) } {{ } k terms of submanifolds in Riemannian space forms.

An Algebraic Inequality with Applications to Certain Chen Inequalities

10.1134/S1063772921050073

It is reduced to describing the divergence of two close geodesics in a Riemannian space and is described by the geodesic deviation equation (Jacobi equation) with the curvature along the geodesic line varying randomly.

Mean Square Geodesic Deviation in the Zeldovich Problem on Light Propagation in a Universe with Inhomogeneities

10.1016/j.jmaa.2020.124712

The last two ones are the weighted Ricci curvatures which also play important roles in weighted Riemannian spaces.

Generalizations of Bonnet-Myers theorem on Finsler manifolds

10.1109/TCDS.2021.3082648

The method uses covariance matrix to represent data feature, and achieves data alignment by rotating the symmetric positive definite (SPD) matrix in Riemannian space.

Transfer Learning: Rotation Alignment with Riemannian Mean for Brain-Computer Interfaces and Wheelchair Control

10.31319/2519-2884.37.2020.14

As for "solid matter", for the compatibility of equations of the gravitational field, it is necessary that particles of dust matter move along geodesics of Riemannian space, which describes the gravitational field.

LAWS OF MOTION IN THE FRAME THEORIES OF GRAVITY

10.1007/s00022-021-00590-7

In the present study, we derive the generalized Wintgen inequality for some submanifolds in metallic Riemannian space forms.

Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms

10.1016/J.GEOMPHYS.2021.104223

spaces) are defined as those homogeneous Riemannian spaces ( M = G ∕ H , g ) whose geodesics are orbits of one-parameter subgroups of G.

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

10.1103/PhysRevD.104.044031

As it is well known, geodesic curves arise as trajectories of structureless test bodies in Riemannian spacetimes with the metric gij as the gravitational field potential, that determines the metric-compatible Christoffel connection Γ̃ij k = 1 2g kl (∂igjl + ∂jgil − ∂lgij).

Demystifying autoparallels in alternative gravity

10.1109/TNNLS.2020.2978514

Empirical investigations on synthetic data and real-world motor imagery EEG data demonstrate that the performance of the proposed generalized learning Riemannian space quantization can significantly outperform the Euclidean GLVQ, generalized relevance LVQ (GRLVQ), and generalized matrix LVQ (GMLVQ).

Generalized Learning Riemannian Space Quantization: A Case Study on Riemannian Manifold of SPD Matrices

If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c / 4 , carrying an indefinite c-Sasakian structure.

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

10.1007/s00521-020-05499-x

Pairwise distances in this Riemannian space, calculated along geodesic paths, can be used to generate a similarity map of the data.

Music genre profiling based on Fisher manifolds and Probabilistic Quantum Clustering

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces.

Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

10.1088/1361-6382/ac274d

We provide a general relativistic analysis of these potentials, by deriving their wave equations in an arbitrary Riemannian spacetime containing a generalised imperfect fluid.

Electromagnetic potentials in curved spacetimes

10.1016/j.optlaseng.2020.106499

The Unitary learning is a Backpropagation serving to unitary weights update through the gradient translation from Euclidean to Riemannian space.

Unitary learning for diffractive deep neural network

10.1007/S10711-021-00639-6

spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G.

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

10.1109/MCG.2021.3069948

Built upon the proposed representation, the 2-D caricature expression extrapolation process can be controlled by the 3-D model reconstructed from the input 2-D caricature image and the exaggerated expressions of the caricature images generated based on the extrapolated expression of a 3-D model that is robust to facial poses in the Kendall shape space; this 3-D model can be calculated with tools such as exponential mapping in Riemannian space.

Caricature Expression Extrapolation Based on Kendall Shape Space Theory

10.1016/j.ifacol.2021.06.164

We show an equivalence between the facts that this system admits three quadratic functionally independent first integrals, the optimal controls are elliptic functions, and the sub-Riemannian space is symmetric.

Elliptic Optimal Control and Symmetric Sub-Riemannian Spaces

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms.

A New Algebraic Inequality and Some Applications in Submanifold Theory

10.33774/coe-2021-qm1l5

“Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time.

Quantity in Quantum Mechanics and the Quantity of Quantum Information

10.15673/TMGC.V14I1.1940

The paper contains necessary conditions allowing to reduce matrix tensors of pseudo-Riemannian spaces to special forms called semi-reducible, under assumption that the tensor defining tensor characteristic of semireducibility spaces, is idempotent.

Special semi-reducible pseudo-Riemannian spaces

10.1142/s0219887821502285

It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space.

Geometric theory of non-regular separation of variables and the bi-Helmholtz equation

10.52526/25792776-2021.68.1-38

Using a way of separating the spectral shifts into infinitesimally displaced `relative´ spectral bins and sum over them, we overcome the ambiguity of the parallel transport of four-velocity, in order to give an unique definition of the so-called kinetic relative velocity of luminous source as measured along the observer’s line-of-sight in a generic pseudo-Riemannian space-time.

Unique definition of relative speed along the line of sight of a luminous object in a Riemannian space-time

10.1109/jstsp.2021.3111438

Two efficient algorithms are developed to solve this non-convex problem on both the Euclidean and Riemannian spaces.

Dual-Functional Radar-Communication Waveform Design: A Symbol-Level Precoding Approach

Some properties of curvature of Riemannian spaces.

Some properties of curvature of Riemannian spaces

In this study we prove that the projective collineations of a $\left( n+1\right) \,$% -dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the $n$-dimensional subspace.

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

10.1007/S10455-020-09751-4

Given a homogeneous pseudo-Riemannian space $$(G/H,\langle \ , \ \rangle),$$ a geodesic $$\gamma :I\rightarrow G/H$$ is said to be two-step homogeneous if it admits a parametrization $$t=\phi (s)$$ (s affine parameter) and vectors X, Y in the Lie algebra $${\mathfrak{g}}$$ , such that $$\gamma (t)=\exp (tX)\exp (tY)\cdot o$$ , for all $$t\in \phi (I)$$.

Two-step homogeneous geodesics in pseudo-Riemannian manifolds

10.1007/978-3-030-19642-4_30

The Fisher information metric defines a Riemannian space where distances reflect similarity with respect to a given probability distribution.

Classifying and Grouping Mammography Images into Communities Using Fisher Information Networks to Assist the Diagnosis of Breast Cancer

10.1007/s00009-019-1408-9

We investigate the relationship of Ricci form, exponential tension field and tension field of the Gauss map of an immersion from a Riemannian manifold into a Riemannian space form.

Exponentially Harmonic Maps, Gauss Maps and Gauss Sections

10.1108/HFF-09-2018-0498

Equally distributing the error among all elements in a suitably defined Riemannian space yields highly anisotropic grids that feature well-resolved shock waves.

A finite element solver for hypersonic flows in thermo-chemical non-equilibrium, Part I

10.1007/S40840-017-0556-Y

In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds.

Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

10.1016/j.jmaa.2020.124081

In this paper, we consider surfaces in 4--dimensional pseudo--Riemannian space--forms with index 2.

Quasi--minimal surfaces of pseudo--Riemannian space forms with positive relative nullity.

10.5507/PRF.19.24455358

Concretely, the monograph treats the following: basic concepts of topological spaces, the theory of manifolds with affine connection (particularly, the problem of semigeodesic coordinates), Riemannian and Kahler manifolds (reconstruction of a metric, equidistant spaces, variational problems in Riemannian spaces, SO(3)-structure as a model of statistical manifolds, decomposition of tensors), the theory of differentiable mappings and transformations of manifolds (the problem of metrization of affine connection, harmonic diffeomorphisms), conformal mappings and transformations (especially conformal mappings onto Einstein spaces, conformal transformations of Riemannian manifolds), geodesic mappings (GM; especially geodesic equivalence of a manifold with affine connection to an equiaffine manifold), GM onto Riemannian manifolds, GM between Riemannian manifolds (GM of equidistant spaces, GM of Vn(B) spaces, its field of symmetric linear endomorphisms), GM of special spaces, particularly Einstein, Kahler, pseudosymmetric manifolds and their generalizations, global geodesic mappings and deformations, GM between Riemannian manifolds of different dimensions, global GM, geodesic deformations of hypersurfaces in Riemannian spaces, some applications of GM to general relativity, namely three invariant classes of the Einstein equations and geodesic mappings, F-planar mappings of spaces with affine connection, holomorphically projective mappings (HPM) of Kahler manifolds (fundamental equations of HPM, HPM of special Kahler manifolds, HPM of parabolic Kahler manifolds, almost geodesic mappings, which generalize geodesic mappings, Riemann-Finsler spaces and their geodesic mappings, geodesic mappings of Berwald spaces onto Riemannian spaces.

Differential Geometry of Special Mappings

The distance duality relation (DDR) is valid in Riemannian spacetime.

Testing the anisotropy of the Universe with the distance duality relation

10.1007/978-3-030-27836-6_6

The physical space-time in general relativity is the four-dimensional pseudo-Riemannian space V4 with the metric signature (+, +, +, −).

Exact Solutions of the Nonlinear Spinor Equations

10.1007/JHEP10(2019)251

ABSTRACTWe present a new consistent truncation of D = 11 supergravity to D = 4 N$$\mathcal{N}$$ = 2 minimal gauged supergravity, on the seven-dimensional internal Riemannian space corresponding to the most general class of D = 11 solutions with an AdS4 factor and N$$\mathcal{N}$$ = 2 supersymmetry.

Minimal D = 4 N$$\mathcal{N}$$ = 2 supergravity from D = 11: an M-theory free lunch

10.1109/ACCESS.2019.2891914

Because CSCM-GW is based on the covariance matrix which belongs to Riemannian space, it has the high computational cost in the recognition phase.

CDF Space Covariance Matrix of Gabor Wavelet With Convolutional Neural Network for Texture Recognition

10.1007/S13366-019-00459-6

They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric (Theorems  3 and 4 ).

On compatible linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds

10.1007/978-3-030-30645-8_58

Experiments were performed to compare the use of Euclidean distance in a vectorial space and a geodesic distance in the Riemannian space of symmetric positive definite matrices.

Analysis of Dynamic Brain Connectivity Through Geodesic Clustering

10.1140/epjc/s10052-019-7310-6

We propose a simple modified gravity model without any initial matter fields in terms of several alternative non-Riemannian spacetime volume elements within the metric (second order) formalism.

Dynamically Generated Inflation from Non-Riemannian Volume Forms

10.1007/S40840-017-0509-5

Also, we consider special almost geodesic mappings of the second type between Eisenhart’s generalized Riemannian spaces as well as between generalized classical (elliptic) and hyperbolic Kähler spaces.

Special Almost Geodesic Mappings of the Second Type Between Generalized Riemannian Spaces

10.14258/IZVASU(2019)1-13

Chibrikova claims that if there is a Weyl tensor with the non-zero squared length for a manifold of dimension n ≥ 4, then a locally homogeneous space can be obtained from a locally conformally homogeneous (pseudo)Riemannian space by means of a conformal deformation.

ИЗОТРОПНЫЙ ТЕНЗОР ВЕЙЛЯ НА ЧЕТЫРЕХМЕРНЫХ ЛОКАЛЬНО ОДНОРОДНЫХ ПСЕВДОРИМАНОВЫХ МНОГООБРАЗИЯХ

10.1007/978-3-030-22643-5_13

In this paper, we shown how to transform the facial expression space to Euclidean space in a way to preserve geometry such as distances and angles in the Riemannian space.

Transform Facial Expression Space to Euclidean Space Using Riemann Normal Coordinates and Its Applications

For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn.

Infinitesimal conformal transformations in the Riemannian space of the second approximation for a space of non-zero constant curvature

10.1007/978-3-030-27836-6_2

Let us recall the basic elementary data concerning Riemannian spaces.

Spinor Fields in a Riemannian Space

10.4028/www.scientific.net/msf.970.276

The paper shows, it is necessary to develop non-Euclidean approach to the crystal’s internal geometry and consider, in consecutive order, the question of the four-dimentional Riemannian space into three-dimentional Eucliden space interpretation (RE interpretation).

Simulation of Structural Transformations in Modified Near-Surface Layers of Crystals

10.1016/J.CJPH.2019.03.010

By exploiting the null tetrad formalism of Jogia and Griffiths and the technique of differential forms on a non-Riemannian space-time, non-static conformally flat, Petrov-type D, spherically symmetric solutions of the Einstein–Cartan field equations; when Weyssenhoff fluid is the source of curvature and spin, are obtained.

Non-static conformally flat spherically symmetric space-times in Einstein–Cartan theory

10.7546/GIQ-20-2019-255-265

Then, we obtained a necessary and sufficient condition for the existence of biconservative quasi-minimal immersions into a four dimensional semi-Riemannian space form of index two with non-zero sectional curvatures.

On Quasi-Minimal Isometric Immersions into Non-Flat Semi Riemannian Space Forms of Index Two

10.1016/J.HM.2019.04.002

In the paper he argued against a naive generalization of analogical reasoning valid for curves and surfaces in three-dimensional Euclidean space to the case of higher-dimensional curved Riemannian spaces.

Václav Hlavatý on intuition in Riemannian space

10.1109/ICDM.2019.00093

In this paper, we extend the learning of adversarial examples to the more general Riemannian space over DNNs.

Generalized Adversarial Training in Riemannian Space

10.1007/978-3-030-22368-7_14

Geodesic curves in this Riemannian space of images give rise to morphing transitions.

Time Discrete Geodesics in Deep Feature Spaces for Image Morphing

10.26512/e-bfis.v8i1.23241

The fundamental interaction of gravitation is very well described by Einstein's General Relativity in a Riemannian spacetime metric, but General Relativity has been over time a gravitational field theory apart from the Standard Model.

Introduction to Gauge Theory of Gravitation

10.1007/S11182-019-01614-1

A possibility of simulating the structure of real nanomaterials in the elliptical Riemannian space is discussed.

Structural Organization of Nanocomposite Crystals

In this work we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension.

Barycentric interpolation on Riemannian and semi-Riemannian spaces

10.23919/EUSIPCO.2019.8903175

This method relied on a multiresolution classifier that tackled class overlaps by using the Riemannian geometry of the RCDs of the multiscale patches of every multichannel image and reducing the dimensionality of these RCDs through a novel method that incorporated the intra-and inter-class neighborhoods of the RCDs in the Riemannian space and the spatial and hierarchical relationships between their corresponding patches.

Spatial and Hierarchical Riemannian Dimensionality Reduction and Dictionary Learning for Segmenting Multichannel Images

10.1142/S0218271819440097

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite.

A measure for quantum paths, gravity and spacetime microstructure

10.1109/IJCNN.2019.8852367

In this paper, we consider optimizing a smooth, convex, lower semicontinuous function in Riemannian space with constraints.

A Riemannian Primal-dual Algorithm Based on Proximal Operator and its Application in Metric Learning

10.1007/s10711-020-00542-6

We study three different topologies on the moduli space $$\mathcal {H}^\mathrm{loc}_m$$ H m loc of equivariant local isometry classes of m -dimensional locally homogeneous Riemannian spaces.

Convergence of locally homogeneous spaces

We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces.

On Conformal and Concircular Diffeomorphisms of Eisenhart’s Generalized Riemannian Spaces

10.1140/EPJC/S10052-019-7044-5

Distinctive features of this model are the absence of a dimensional parameter for the nonlinearity description and a very simple form of the dominant energy condition, which can easily be verified in an arbitrary pseudo-Riemannian space-time with the consequent constraints on the model parameters.

Compact objects in conformal nonlinear electrodynamics

10.21136/CMJ.2019.0562-17

The purpose of this article is to obtain sharp estimates for the first eigenvalue of the stability operator of constant mean curvature closed hypersurfaces immersed into locally symmetric Riemannian spaces satisfying suitable curvature conditions (which includes, in particular, a Riemannian space with constant sectional curvature).

Sharp eigenvalue estimates of closed H-hypersurfaces in locally symmetric spaces

Here we apply the general scheme for description of the mechanics of infinitesimal bodies in the Riemannian spaces to the examples of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloons.

Mechanics of the Infinitesimal Gyroscopes on the Mylar Balloons and Their Action-Angle Analysis.

10.1109/SMC.2019.8914414

Methods: We propose two approaches to enhance the performance of a state-of-the-art Riemannian space transfer learning (TL) algorithm: 1) trials selection, which resamples trials from the auxiliary subjects so that they become more consistent with those of the new subject; and, 2) channel selection, which reduces the number of channels and hence makes the Riemannian space computations more accurate and efficient.

Channel and Trials Selection for Reducing Covariate Shift in EEG-based Brain-Computer Interfaces

In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces.

Conformal and Geodesic Mappings onto Some Special Spaces