Renormalization and conformal properties of sigma models on Riemannian space with torsion.

Symmetry Reduction for Nonlinear wave Equations in Riemannian and Pseudo-Riemannian Spaces

We study two and three-point tree-level amplitudes of open strings. These amplitudes are reduced from higher-point correlation functions by using mostly BRST exact operators for gauge fixing. For simplicity, we focus on an open string tachyon. The two-point amplitude of open string tachyons is reduced from a three-point function of two tachyon vertex operators and one mostly BRST exact operator. Similarly the three-point amplitude is from a four-point function of three tachyon vertex operators and one mostly BRST exact operator. One can also obtain the two-point amplitude from the four-point function of two tachyon vertex operators and two mostly BRST exact operators. In these derivation from four-point functions, moduli integrals are significant. We discuss the overall signs of amplitudes which are indefinite in this formalism.

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W. Klingenberg. The key point in the study of Finsler manifolds is the non-symmetric property of its distance functions. We discuss fundamental results on the cut and conjugate loci of Finsler manifolds and note the differences between Riemannian and Finsler manifolds in these respects. The topological and differential structures on Riemannian manifolds, in the presence of convex functions, has been an active field of research in the second half of twentieth century. We discuss some results on Riemannian manifolds with convex functions and their recently proved analogues in the field of Finsler manifolds.

Some unified theories on the gravitational and electromagnetic fields are researched. We investigate mainly two new geometric unified theories. A method is that the gravitational field and the source-free electromagnetic field can be unified by the equations Rklmi = κTklmi* in the Riemannian geometry, both are contractions of im and ik, respectively. If Rklmi = κTklmi* =constant, it will be equivalent to the Yang’s gravitational equations Rkm;l –Rkl;m = 0, which include Rlm= 0. From Rlm= 0 we can obtain the Lorentz equations of motion, the first system and second source-free system of Maxwell’s field equations. This unification can be included in the gauge theory, and the unified gauge group is SL(2,C) × U(1)=GL(2,C), which is just the same as the gauge group of the Riemannian manifold. Another unified method on the general nonsymmetric metric field with high-dimensional space-time and its matrix representations are analyzed mathematically. Further, the general unified theory of five-dimensional space-time combined quantum theory and four interactions is researched. Some possible unification ways on the gravitational and electromagnetic fields are discussed. The general matrix and various corresponding theories may decompose to a sum of symmetry and antisymmetry. Moreover, we proposed an imaginative representation on the ten dimensional space-time.

Abstract. Let G be a compact connected semisimple Lie group, g theLie algebra of G, g the canonical metric (the biinvariant Riemannianmetric which is induced from the Killing form of g), and r be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection r in the tangent bundle TG over (G;g) is a Yang-Millsconnection. 1. IntroductionThe problem of nding metrics and connections which are critical pointsof some functional plays an important role in global analysis and Riemann-ian geometry. A Yang-Mills connection is a critical point of the Yang-MillsfunctionalYM(D) =12Z M kR D k 2 v g (1:1)on the space C E of all connections in a smooth vector bundle Eover a closed(compact and connected) Riemannian manifold (M;g), where R D is the cur-vature of D2C E . Equivalently, Dis a Yang-Mills connection if it satis es theYang-Mills equation (cf. [1, 5, 6]) D R D = 0; (1:2)(the Euler-Lagrange equations of the variational principle associated with (1.1)).If Dis a connection in a vector bundle Ewith bundle metric hover a Riemann-ian manifold (M;g), then the connection D given byh(D

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<p>This is an obsolete version of the article - https://doi.org/10.5281/zenodo.4040036</p> <p> </p> <p>The first main work of this paper is to generalize intrinsic geometry. (1) Riemannian manifold is generalized to geometrical manifold. (2) The expression of Erlangen program is improved, and the concept of intrinsic geometry is generalized, so that Riemannian intrinsic geometry which is based on the first fundamental form becomes a subgeometry of the generalized intrinsic geometry. The Riemannian geometry is thereby incorporated into the geometrical framework of improved Erlangen program. (3) The important concept of simple connection is discovered, which reflects more intrinsic properties of manifold than Levi-Civita connection.</p> <p> </p> <p>The second main work of this paper is to apply the generalized intrinsic geometry to Hilbert's 6th problem at the most basic level. (1) It starts from an axiom and makes key principles, postulates and artificially introduced equations of fundamental physics all turned into theorems which automatically hold in intrinsic geometrical theory. (2) Intrinsic geometry makes gravitational field and gauge field unified essentially. Intrinsic geometry of external space describes gravitational field, and intrinsic geometry of internal space describes typical gauge field. They are unified into intrinsic geometry. (3) Intrinsic geometry makes gravitational theory and quantum mechanics have the same view of time and space and unified description of evolution.</p>

The geometry of Riemannian symmetric spaces is really richer than that of Riemannian homogeneous spaces. Nevertheless, there exists a large literature of special classes of homogeneous Riemannian manifolds with an important list of features which are typical for a Riemannian symmetric space. Normal homogeneous spaces, naturally reductive homogeneous spaces or g. o. spaces are some interesting examples of these classes of spaces where, in particular, the Jacobi equation can be also written as a differential equation with constant coefficients and the osculating rank of the Jacobi operator is constant. Compact rank one symmetric spaces are among the very few manifolds that are known to admit metrics with positive sectional curvature. In fact, there exist only three non-symmetric (simply-connected) normal homogeneous spaces with positive curvature: V 1 = S p (2)/SU(2), V 2 = SU(5)/(S p (2) × S 1 ), given by M. Berger and the Wilking’s example V 3 = (SU(3) SO(3))/U • (2). Here, we show some geometric properties of all these spaces, properties related with the existence of isotropic Jacobi fields and the determination of the constant osculating rank of the Jacobi operator. It provides different way to ”measure” of how they are so close or not to the class of compact rank one symmetric spaces.

Depth images captured by consumer depth sensors like ToF Cameras or Microsoft Kinect are often noisy and incomplete. Most existing methods recover missing depth values from low quality measurements using information in the corresponding color images. However, the performance of such methods is susceptible when color image is noisy or correlation between RGB-D is weak. This paper presents a depth map enhancement algorithm based on Riemannian Geometry that performs depth map de-noising and completion simultaneously. The algorithm is based on the observation that similar RGB-D patches lie in a very low-dimensional subspace over the Riemannian quotient manifold of varying-rank matrices. The similar RGB-D patches are assembled into a matrix and optimization is performed on the search space of this quotient manifold with Kronecker product trace norm penalty. The proposed convex optimization problem on a special quotient manifold essentially captures the underlying structure in the color and depth patches. This enables robust depth refinement against noise or weak correlation between RGB-D data. This non-Euclidean approach with Kronecker product trace-norm constraints and cones in the non-linear matrix spaces provide a proper geometric framework to perform optimization. This formulates depth map enhancement as a matrix completion problem in the product space of Riemannian manifolds. This Riemannian submersion automatically handles ranks that change over matrices, and ensures guaranteed convergence over constructed manifold. The experiments on public benchmarks RGB-D images show that proposed method can effectively enhance depth maps.

We consider the problem of constructing scalar particle wave equations in Riemannian spaces with external gauge fields whose symmetry group is the group of motions of the Riemannian space.

The present contribution shows that warped Riemannian metrics, a class of Riemannian metrics which play a prominent role in Riemannian geometry, are also of fundamental importance in information geometry. Precisely, the starting point is a new theorem, which states that the Rao–Fisher information metric of any location-scale model, defined on a Riemannian manifold, is a warped Riemannian metric, whenever this model is invariant under the action of some Lie group. This theorem is a valuable tool in finding the expression of the Rao–Fisher information metric of location-scale models defined on high-dimensional Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by only two functions of a single variable, irrespective of the dimension of the underlying Riemannian manifold. Starting from this theorem, several original results are obtained. The expression of the Rao–Fisher information metric of the Riemannian Gaussian model is provided, for the first time in the literature. A generalised definition of the Mahalanobis distance is introduced, which is applicable to any location-scale model defined on a Riemannian manifold. The solution of the geodesic equation, as well as an explicit construction of Riemannian Brownian motion, are obtained, for any Rao–Fisher information metric defined in terms of warped Riemannian metrics. Finally, using a mixture of analytical and numerical computations, it is shown that the parameter space of the von Mises–Fisher model of n-dimensional directional data, when equipped with its Rao–Fisher information metric, becomes a Hadamard manifold, a simply-connected complete Riemannian manifold of negative sectional curvature, for \(n = 2,\ldots ,8\). Hopefully, in upcoming work, this will be proved for any value of n.

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.

In its most general form, the recognition problem in riemannian geometry asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold. Optimally one wants to recognize a manifold having made as few measurements as possible. Many results in riemannian geometry, including pinching theorems, can be viewed this way. Here we are only interested in measurements that assign real numbers to each (complete) riemannian manifold. Typical examples of such invariants are diameter, volume, curvature bounds, etc. When viewing one or several invariants, I = (II ' . .. ,II)' of this type as a map on a suitable class, L , of riemannian manifolds, the following problems pose themselves:

Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, i.e., how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low-dimensional space. Experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.