Search

This paper advocates Riemannian multi-manifold modeling for network-wide time-series analysis: Dynamic brain-network data yield features which are viewed as points in or close to a union of a finite number of submanifolds of a Riemannian manifold. Distinguishing disparate time series amounts then to clustering multiple Riemannian submanifolds. To this end, two feature-generation schemes for network-wide dynamic time series are put forth. The first one is motivated by Granger-causality arguments and uses an auto-regressive moving average model to map low-rank linear vector subspaces, spanned by column vectors of observability matrices, to points into the Grassmann manifold. The second one utilizes (non-linear) dependencies among network nodes by introducing kernel-based partial correlations to generate points in the manifold of positive-definite matrices. Capitalizing on recently developed research on Riemannian-submanifold clustering, an algorithm is provided to differentiate time series based on their Riemannian-geometry properties. Extensive numerical tests on synthetic and real fMRI data demonstrate that the proposed framework outperforms classical and state-of-the-art techniques in clustering brain-network states/structures.

We prove that a Riemannian manifold admitting a metric connection with totally skew-symmetric torsion and reducible holonomy is locally reducible, provided it has nonpositive sectional curvature.

In the present paper, we introduce slant Riemannian maps from an almost contact manifold to Riemannian manifolds. We obtain the existence condition of slant Riemannian maps from an almost contact manifold to Riemannian manifolds. Moreover, we find the necessary and sufficient condition for slant Riemannian map to be totally geodesic and investigate the harmonicity of slant Riemannian maps from Sasakian manifold to Riemannian manifolds. Finally, we obtain a decomposition theorem for the total manifolds and also provide some examples of such maps.

In this paper, we study Riemannian, anti-invariant and Lagrangian submersions from locally product Riemannian manifolds onto Riemannian manifolds. We first give a characterization theorem for Riemannian submersions. It is proved that the fibers of a Lagrangian submersion are always totally geodesic. We also consider the first variational formula of anti-invariant Riemannian submersions and give a new condition for the harmonicity of such submersions.

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.

A pair of points x, y in a Riemannian manifold ( M , g ) is said to be secure if there exists a finite set of points intercepting every geodesic segment joining x to y. Given any conformal equivalence class C of Riemannian metrics on a closed manifold M of dimension at least two and given any pair of points x, y in M, there exists a dense G δ set C ′ ⊂ C such that x and y are not secure for every metric g in C ′ .

In this paper, an upper subderivative of a lower semicontinuous function on a Riemannian manifold is introduced. Then, an approximate mean value theorem for the upper subderivative on a Hadamard manifold is presented. Moreover, the results are used for characterization of convex functions on Riemannian manifolds.

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions. As a consequence we can characterize sub-Riemannian geodesics as the horizontal lifts of projections of Riemannian geodesics., 12 pages

Let be the n-dimensional pseudo-Euclidean space of index p and M(n, p) the group of all transformations of generated by pseudo-orthogonal transformations and parallel translations. We describe the system of generators of the differential field of all M(n, p)-invariant differential rational functions of a map of an open subset . Using this result, we prove analogues of the Bonnet theorem for immersions of an n-dimensional C∞-manifold J in . These analogues are given in terms of the pseudo-Riemannian metric, the volume form, and the connection on J induced by the immersion of J in .

In this paper, we present a new trust region algorithm on any compact Riemannian manifolds using subspace techniques. The global convergence of the method is proved and local \(d+1\)-step superlinear convergence of the algorithm is presented, where d is the dimension of the Riemannian manifold. Our numerical results show that the proposed subspace algorithm is competitive to some recent developed methods, such as the LRTR-SR1 method, the LRTR-BFGS method, the Riemannian CG method.

In this paper, we study pseudo-slant submanifolds of a locally decom- posable Riemannian manifold. We give necessary and suffcient conditions for distributions which are involued in the definition of pseudo-slant sub- manifold to be integrable. We search these type submanifolds with parallel canonical structures and we obtain some new results.

In this short paper, as applications of the well-known generalized maximum principle of Omori–Yau, we obtain new extensions of a classical theorem due to Moser [8] . More precisely, under suitable constraints on the norm of the gradient of the smooth function u that defines an entire CMC graph Σ ( u ) constructed over a fiber M n of a Riemannian product space of the type R × M n , we show that u must actually be constant.

In this paper, we define a special new family of metrics which rescale the horizontal part by a nonzero differentiable function on the tangent bundle over a Riemannian manifold. We investigate curvature properties of the Levi-Civita connection and another metric connection of the new Riemannian metric.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of K\"ahler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.

The object of the present paper is to characterize a Riemannian manifold admitting a type of semi-symmetric non-metric connection.

The paper relates to the circle of problems concerning the connection between the conformal type of a Riemannian manifold and the canonical form of its isoperimetric function. Two special examples of 2-manifolds are constructed, which explain the meaning, role and importance of the conditions involved in the criterion, previously obtained by the author, which decides whether a noncompact Riemannian n-manifold is conformally parabolic. Bibliography: 8 titles.

The minimal hypersurface equation for a graph in a Riemannian manifold which admits a nowhere zero Killing vector field, whose orthogonal distribution is integrable, is studied. New uniqueness results for the entire solutions of this equation on a compact Riemannian manifold of arbitrary dimension are given. In particular, new Bernstein theorems are proved.

Let M be a differentiable manifold equipped with a Riemannian metric tensor. If we apply translations in the tangent spaces to the Riemannian unit balls such that the translated bodies are keeping the origin in their interiors then they are working as unit balls for a Randers manifold. The unit balls allow us to measure the length of tangent vectors with the help of the induced Minkowski functionals. Analytically these Minkowski functionals are coming from a Riemannian metric tensor by using one-form perturbation in the tangent spaces. Manifolds equipped with a smoothly varying family of Minkowski functionals are called Finsler manifolds. The one-form perturbation of a Riemannian metric in the tangent spaces results in the class of Randers manifolds introduced by G. Randers in 1941. They occur naturally in physical applications related to electron optics, navigation problems or the Lagrangian of relativistic electrons. In this paper we are interested in Randers manifolds with semi-symmetric compatible linear connections. Compatibility means that the parallel transports preserve the length of tangent vectors with respect to the perturbed metric. If the torsion is decomposable in a special way then we speak about a semi-symmetric linear connection. Up to local isometries we characterize Riemannian manifolds admitting a one-form perturbation such that the resulting Randers manifold has a compatible semi-symmetric linear connection. As a paraphrase of our previous work (Vincze, 2006) communicated by Professor J.J. Duistermaat we present an existence theorem of generalized Berwald manifolds with semi-symmetric compatible linear connections. The terminology goes back to M. Matsumoto, M. Hashiguchi and S. Bacso’s work (Bacso et al., 1997).

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.

International audience; This paper introduces a novel algorithm for the online estimate of the Riemannian mixture model parameters. This new approach counts on Riemannian geometry concepts to extend the well-known Tit-terington approach for the online estimate of mixture model parameters in the Euclidean case to the Riemannian manifolds. Here, Riemannian mixtures in the Riemannian manifold of Symmetric Positive Definite (SPD) matrices are analyzed in details, even if the method is well suited for other manifolds.

We introduce semi-slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of semi-slant immersions, invariant Riemannian maps, anti-invariant Riemannian maps and slant Riemannian maps. We obtain characterizations, investigate the harmonicity of such maps and find necessary and sufficient conditions for semi-slant Riemannian maps to be totally geodesic. Then we relate the notion of semi-slant Riemannian maps to the notion of pseudo-horizontally weakly conformal maps, which are useful for proving various complex-analytic properties of stable harmonic maps from complex projective space and give many examples of such maps.

In this paper, the applicability of two optimization algorithms in the context of Riemannian manifolds is investigated. More precisely, the steepest descent method and the nonlinear conjugate gradient method are used to solve the Shape-From-Shading problem in a shape space of triangular meshes. For this reason, appropriate Riemannian metrics are introduced in this shape space. Instead of taking steps along straight lines, propagation along geodesics with respect to the Riemannian metric is employed. This requires to formulate the geodesic equation and the equation of parallel translation in the shape space of triangular meshes. In general, the proposed geodesic optimization algorithms outperform the standard steepest descent method concerning various quality measures and the visual impression of the reconstructed surface. In addition, it is shown that an appropriately chosen Riemannian metric can guide the optimization process towards a desired class of surfaces.

On a Lie group G, the restrictions of a homomorphism h:(R,+)→(G,∘) to intervals [a,b] are geodesic according to the left- or right-invariant connection. If this affine connection is torsion free, G is a Riemannian manifold. On Riemannian manifolds in general, it's difficult to compute geodesics (i.e. locally shortest lines) between two points, which are far away from each other. But any curve connecting the two points can be shortened by using a ruler, which allows to construct short geodesics:In normed vector spaces, we look at the midpoint method and the reduced method, which shorten the curve iterative. Both let converge a curve to the geodesic: the straight line between starting point and end point of the curve. On Riemannian manifolds, we get a weaker result: The reduced method can be generalized to complete Riemannian manifolds and on some of them – as for example the hyperbolic spaces – the reduced method lets converge each curve to a geodesic between its starting point and end point.

This paper introduces mathematical morphology operators for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonical quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonical case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We show also how recent results on Lasry-Lions regularization can be used for non-smooth image filtering based on morphological Riemannian operators. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing real-valued 3D surfaces.

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

We discuss the use of the Newton method in the computation of max(p → Ep [f]), where p belongs to a statistical exponential family on a finite state space. In a number of papers, the authors have applied first order search methods based on information geometry. Second order methods have been widely used in optimization on manifolds, e.g., matrix manifolds, but appear to be new in statistical manifolds. These methods require the computation of the Riemannian Hessian in a statistical manifold. We use a non-parametric formulation of information geometry in view of further applications in the continuous state space cases, where the construction of a proper Riemannian structure is still an open problem.

We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

The well-known symmetric rank-one trust-region method--where the Hessian approximation is generated by the symmetric rank-one update--is generalized to the problem of minimizing a real-valued function over a $$d$$ d -dimensional Riemannian manifold. The generalization relies on basic differential-geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on the more recent notions of (first-order) retraction and vector transport. The new method, called RTR-SR1, is shown to converge globally and $$d+1$$ d + 1 -step q-superlinearly to stationary points of the objective function. A limited-memory version, referred to as LRTR-SR1, is also introduced. In this context, novel efficient strategies are presented to construct a vector transport on a submanifold of a Euclidean space. Numerical experiments--Rayleigh quotient minimization on the sphere and a joint diagonalization problem on the Stiefel manifold--illustrate the value of the new methods.

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Lowner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.

B. Sahin [12] introduced the notion of semi-invariant Riemannian submersions as a generalization of anti-invariant Riemmanian submersions [11]. As a generalization to semi-invariant Riemannian submersions we introduce the notion of generic submersion from an almost Hermitian manifold onto a Riemannian manifold and investigate the geometry of foliations which arise from the definition of a generic Riemannian submersion and find necessary and sufficient condition for total manifold to be a generic product manifold. We also find necessary and sufficient conditions for a generic submersion to be totally geodesic.

In this paper we will show that for every positive integer k and each pair of points p, q ∈ M, where M is a Riemannian manifold diffeomorphic to the 2-dimensional sphere, there always exist at least k geodesics connecting p and q of length at most 22kd, where d is the diameter of M.

The equivalence problem of curves with values in a Riemannian manifold is solved. The domain of validity of Frenet’s theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold, new invariants must thus be added. There are two important generic classes of curves: namely Frenet curves and a new class called curves “in normal position”. They coincide in dimensions \(\le 4\) only. A sharp bound for asymptotic stability of differential invariants is obtained, the complete systems of invariants are characterized, and a procedure of generation is presented.

Let $M$ be a compact smooth Riemannian manifold of finite dimension $n+1$ with boundary $\partial M$and $\partial M$ is a compact $n$-dimensional submanifold of $M$. We show that for generic Riemannian metric $g$, all the critical points of the mean curvature of $\partial M$ are nondegenerate.

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.

In this chapter, we study Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. In section 1, we study holomorphic Riemannian maps as a generalization of holomorphic submersions and obtain a characterization of such maps. In section 2, we investigate anti-invariant Riemannian maps as a generalization of anti-invariant submersion, investigate the geometry of leaves of distributions defined by such maps, and give necessary and sufficient conditions for anti-invariant Riemannian maps to be totally geodesic. We also find necessary and sufficient conditions for the total manifold of anti-invariant Riemannian maps to be an Einstein manifold. In section 3, as a generalization of the semi-invariant submersion, we introduce semi-invariant Riemannian maps, give examples, and obtain the main properties of such maps. In section 4, we introduce generic Riemannian maps from almost Hermitian manifolds to Riemannian manifolds as a generalization of generic submersions and we give examples. We also find new conditions for Riemannian maps to be totally geodesic and harmonic. In section 5, we study slant Riemannian maps as a generalization of slant submersion, give examples, and obtain the harmonicity of such maps. We also find new necessary and sufficient conditions for such maps to be totally geodesic. Moreover, we obtain a decomposition theorem by slant Riemannian maps. In section 6, we define semi-slant Riemannian maps and give examples. In section 7, we introduce hemi-slant Riemannian maps as a generalization of hemi-slant submersions and slant Riemannian maps, give examples, obtain integrability conditions for distribution, and investigate the geometry of these distributions. We also obtain conditions for such maps to be harmonic and totally geodesic, respectively.

In this chapter, we study Riemannian maps between Riemannian manifolds. In section 1, we define Riemannian maps and give the main properties of such maps. In section 2, we obtain Gauss-Weingarten-like formulas and then we obtain Gauss, Codazzi, and Ricci equations along Riemannian maps. In section 3, we find necessary and sufficient conditions for Riemannian maps to be totally geodesic by using the Bochner identity and generalized divergence theorem. In section 4, we introduce umbilical Riemannian maps and pseudoumbilical Riemannian maps, and obtain characterizations of such Riemannian maps. In section 5, we discuss the harmonicity and biharmonicity of Riemannian maps. In section 6, we define Clairaut Riemannian maps, give an example, and obtain a characterization. In section 6, we extend the result of Nomizu-Yano to the Riemannian maps. In section 7, we obtain Chen’s inequality for Riemannian maps. In the last section, we investigate necessary and sufficient conditions for Riemannian maps to have the Einstein property.

In this chapter, we study Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. In section 1, we study invariant Riemannian maps, that is, the image of derivative map is invariant under the almost complex structure of the base manifold. We give examples, investigate the geometry of foliations, and find new conditions for the harmonicity of such maps. In section 2, we study anti-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give several examples, and obtain a method to find anti-invariant Riemannian maps. We also obtain a characterization for umbilical anti-invariant Riemannian maps. In section 3, we study semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of CR-submanifolds. We give examples, obtain the totally geodesicity of such maps, and relate it with PHWC maps. In section 4, we investigate generic Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of generic submanifolds. We give examples and obtain necessary and sufficient conditions for such maps to be totally geodesic and harmonic. In section 5, we introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant submanifolds. We obtain characterizations for such maps and investigate the geometry of maps. In section 6, we study semi-slant Riemannian maps as a generalization of semi-slant submanifolds, give examples, and obtain new characterizations. In section 7, we define hemi-slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of hemi-slant submanifolds and slant Riemannian maps to almost Hermitian manifolds. We give examples, obtain a decomposition theorem and find necessary and sufficient conditions to be a totally geodesic map.

In this chapter, we give brief information about geometric structures which will be used in the following chapters. In addition to the basic concepts, some geometric notions such as symmetry conditions, parallelity conditions, new product structures on manifolds, generalized Einstein manifolds, and biharmonic maps introduced very recently will be presented in this chapter. The chapter consists of six sections. In the first section, the main notions and theorems of Riemannian geometry are reviewed, such as the Riemannian manifold, Riemannian metric, Riemannian connection, curvatures (Riemannian, sectional Ricci, Scalar), derivatives (exterior, covariant, inner), operators (Hessian, divergence, Laplacian), Einstein manifolds and their generalizations, symmetry conditions, and very brief notions from integration. In the second section, we look at vector bundles and related notions. In the third section, we recall basic notions from submanifold theory and distributions on manifolds. In the fourth section, we mention Riemannian submersions, O’Neill’s tensor fields, and curvature relations of Riemannian submersions. In this section, we also recall the notion of horizontally weakly conformal maps, which will be a crucial tool for a characterization of harmonic maps. In the fifth section, we study various product structures including warped product, twisted product, oblique warped product, and convolution product on Riemannian manifolds. We also provide connection relations and curvature expressions of each case. In the last section, we give a general setting for a map between manifolds such as a vector field along a map, a connection along a map, a curvature tensor field along a map, a second fundamental form of a map, a tension field of a map. We also recall harmonic maps and biharmonic maps in detail.

The problem of nonlinear alignment of functions is both fundamental and extremely important in pattern recognition. Most common approaches for alignment, including dynamic time warping (DTW), use penalized-\(\mathbb {L}^2\) minimization that has significant shortcomings, including asymmetry. A recent mathematical framework, based on an elastic Riemannian metric and square-root velocity functions, overcomes these shortcomings. The time warping problem is currently solved using a dynamic programming algorithm (DPA) which relies heavily on dense sampling of functions and can be computationally expensive. Here we present a novel theory (and algorithms) for finding the exact pairwise alignment between functions, which uses an efficient sampling of functions, restricted to their change points. In many cases, the computational cost for matching is reduced by orders of magnitude. We demonstrate the superiority of this method over the DPA using several simulated and real datasets.

We first introduce hemi-slant Riemannian maps and study such Riemannian maps from Kahler manifolds onto Riemannian manifolds. We give necessary and sufficient conditions for the integrability of the distributions which are involved in the definition of the Riemannian map and investigate their leaves. We also obtain harmonicity and totally geodesic conditions for such maps. Then we introduce hemi-slant Riemannian maps from arbitrary Riemannian manifolds to Kahler manifolds and find a decomposition theorem for the image. We also provide examples for both cases.

We show the tensor properties of joint probabilities on a Riemannian parametric manifold. Initially we develop a binary data matrix for parameter measurements of a large number of particles confining in a closed system in order to retrieve the joint probability densities of related parameters. By introducing a new generalized inner product as a multilinear operation on dual basis of parametric space, we extract the set of joint probabilities and prove them to meet contravariant tensor properties in a general Riemannian parametric space. We show these contravariant tensors reduce to classical ordinary partial derivatives definition in ordinary Euclidean parametric space. Finally we prove by a theorem that symmetrized iterative contravariant derivative of cumulative probability function on Riemannian manifold gives the set of joint probabilities in those manifold. We bring some examples for compatibility with physical tensors.

Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.

We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence., 30 pages, 5 figures, Matlab code

This paper investigates the optimal control of multi-joint manipulator from the view of Riemannian geometry. We construct Riemannian manifold equipped with kinetic energy metric about configuration space, and regard constraints work space as submersion submanifold. We define the covariant forms for velocity and acceleration on Riemannian manifold, which make all expressions be independent of the choice of the coordinate systems, and research the relationship between the Euler equation and the geodesic equation, and show the property of preserving energy along geodesic paths. We investigate the hybrid position and force control under constraints, and analyze the corresponding stability based on Lyapunov’s method.

We deal with the Q -curvature problem on a 4-dimensional compact Riemannian manifold ( M , g ) with ∫ M Q g d V g = 8 π 2 and positive Paneitz operator P g . Let Q be a positive smooth function. The question we consider is, when can we find a metric g which is conformal to g , such that Q is just the Q -curvature of g . A sufficient condition to this question is given in this paper.

In this paper we show that a 3-dimensional non-Sasakian contact metric manifold [ M , ( η , ξ , ϕ , g ) ] is a ( κ , μ , ν ) -contact metric manifold with ν = const. , if and only if there exists a Riemannian g-natural metric G ˜ on T 1 M for which ξ : ( M , g ) ↦ ( T 1 M , G ˜ ) is a harmonic map. Furthermore, we give examples of 3-dimensional non-Sasakian contact metric manifolds [ M , ( η , ξ , ϕ , g ) ] such that the corresponding Reeb vector fields ξ : ( M , g ) ↦ ( T 1 M , G ˜ ) are harmonic maps, for suitable Riemannian g-natural metrics G ˜ on T 1 M which are not of Kaluza–Klein type. Finally, we prove that if ( M , g ) is an Einstein manifold and ( η ˜ , ξ ˜ , ϕ ˜ , G ˜ ) a g-natural contact metric structure on T 1 M , then the contact metric manifold [ T 1 M , ( η ˜ , ξ ˜ , ϕ ˜ , G ˜ ) ] is H-contact if and only if ( M , g ) is 2-stein.

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a geodesic "superflow" on the cotangent bundle of the supermanifold. Integral curves of this flow turn out to be in natural bijection with geodesics on the Riemannian supermanifold. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds., 28 pages

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenbock–Lichnerowicz type formula which allows us to apply the classical Bochner–Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.