## What is/are Riemannian Space?

Riemannian Space - 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.^{[1]}“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.

^{[2]}— 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.

^{[3]}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.

^{[4]}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.

^{[5]}By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.

^{[6]}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.

^{[7]}In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve.

^{[8]}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.

^{[9]}, 2) } {{ } k terms of submanifolds in Riemannian space forms.

^{[10]}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.

^{[11]}The last two ones are the weighted Ricci curvatures which also play important roles in weighted Riemannian spaces.

^{[12]}The method uses covariance matrix to represent data feature, and achieves data alignment by rotating the symmetric positive definite (SPD) matrix in Riemannian space.

^{[13]}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.

^{[14]}In the present study, we derive the generalized Wintgen inequality for some submanifolds in metallic Riemannian space forms.

^{[15]}spaces) are defined as those homogeneous Riemannian spaces ( M = G ∕ H , g ) whose geodesics are orbits of one-parameter subgroups of G.

^{[16]}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).

^{[17]}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).

^{[18]}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.

^{[19]}Pairwise distances in this Riemannian space, calculated along geodesic paths, can be used to generate a similarity map of the data.

^{[20]}We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces.

^{[21]}We provide a general relativistic analysis of these potentials, by deriving their wave equations in an arbitrary Riemannian spacetime containing a generalised imperfect fluid.

^{[22]}The Unitary learning is a Backpropagation serving to unitary weights update through the gradient translation from Euclidean to Riemannian space.

^{[23]}spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G.

^{[24]}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.

^{[25]}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.

^{[26]}We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms.

^{[27]}“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.

^{[28]}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.

^{[29]}It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space.

^{[30]}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.

^{[31]}Two efficient algorithms are developed to solve this non-convex problem on both the Euclidean and Riemannian spaces.

^{[32]}Some properties of curvature of Riemannian spaces.

^{[33]}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.

^{[34]}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 $$t\in \phi (I)$$.

^{[35]}The Fisher information metric defines a Riemannian space where distances reflect similarity with respect to a given probability distribution.

^{[36]}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.

^{[37]}Equally distributing the error among all elements in a suitably defined Riemannian space yields highly anisotropic grids that feature well-resolved shock waves.

^{[38]}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.

^{[39]}In this paper, we consider surfaces in 4--dimensional pseudo--Riemannian space--forms with index 2.

^{[40]}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.

^{[41]}The distance duality relation (DDR) is valid in Riemannian spacetime.

^{[42]}The physical space-time in general relativity is the four-dimensional pseudo-Riemannian space V4 with the metric signature (+, +, +, −).

^{[43]}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.

^{[44]}Because CSCM-GW is based on the covariance matrix which belongs to Riemannian space, it has the high computational cost in the recognition phase.

^{[45]}They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric (Theorems 3 and 4 ).

^{[46]}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.

^{[47]}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.

^{[48]}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.

^{[49]}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.

^{[50]}

## Homogeneou Riemannian Space

spaces) are defined as those homogeneous Riemannian spaces ( M = G ∕ H , g ) whose geodesics are orbits of one-parameter subgroups of G.^{[1]}spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G.

^{[2]}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.

^{[3]}

## Generalized Riemannian Space

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.^{[1]}We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces.

^{[2]}

## Learning Riemannian Space

By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.^{[1]}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).

^{[2]}

## riemannian space form

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.^{[1]}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.

^{[2]}In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve.

^{[3]}, 2) } {{ } k terms of submanifolds in Riemannian space forms.

^{[4]}In the present study, we derive the generalized Wintgen inequality for some submanifolds in metallic Riemannian space forms.

^{[5]}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.

^{[6]}We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms.

^{[7]}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.

^{[8]}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.

^{[9]}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.

^{[10]}

## riemannian space vn

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.^{[1]}For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn.

^{[2]}

## riemannian space quantization

By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent.^{[1]}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).

^{[2]}