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Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations, for which a geometric approach was introduced by Christodoulou and Lindblad [11]. Due to the loss of the one more derivative and loss of the symmetry of equations which create significant difficulties in closing the estimates in Sobolev spaces of finite regularity, the geometric approach in [11] needs to be substantially developed and extended by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by “localization technique” using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds$$S^2$$S2andSO(3).

We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fr\'{e}chet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing., Comment: 19 pages

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary, cornered asymptotically hyperbolic manifolds, and proves a theorem of Cartan–Hadamard-type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary.

Let k be a field. We show that the ring of invariants of every exponential map on k[x, y] is generated by the image of x or y under a composition of triangular automorphisms. From this we obtain two well-known theorems on k[x, y] with no restriction on the characteristic of k: the Rentschler–Miyanishi theorem and the Jung–van der Kulk theorem.

Let $\Cal F$ be a singular Riemannian foliation on a compact connected Riemannian manifold $M$. We demonstrate that global foliated vector fields generate a distribution tangent to the strata defined by the closures of leaves of $\Cal F$ and which, in each stratum, is transverse to these closures of leaves.

We define a Lévy process on a smooth manifold M with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of M, driven by a holonomy-invariant Lévy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic Lévy process introduced in Applebaum and Estrade [3]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments.\ud \ud Our main theorem characterizes the class of Lévy processes via their generators on M, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade [32] to smooth manifolds with non-unique geodesics between distinct points.

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato’s Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The key ingredient is the explicit computation of the integral image of the derived Kato’s Euler systems under the dual exponential map. We provide explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.

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 investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.

We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedron from a three-dimensional exterior viewer. The pictures were constructed by computing the exponential map of a polyhedron by cutting and rotating faces into the tangent plane of the bug.

We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemannian manifolds contains a residual set of the metrics on a given smooth manifold of dimension $3$.

The rich information in the power spectral density (PSD) matrix of a received signal can be extracted in different ways for various purposes in signal processing. Care, however, must be taken when distances between PSD matrices are considered because these matrices form a Riemannian manifold in the signal space. In this paper, two Riemannian distances (RD) are introduced for the measurement of distances between PSD matrices on the manifold. The principle by which the geodesics on the manifold can be lifted to a Euclidean subspace isometric with the tangent space of the manifold is also explained and emphasized. This leads to the concept that any optimization involving the RD on the manifold can be equivalently performed in this Euclidean subspace. The application of this principle is illustrated by the development of iterative algorithms to find the Riemannian means and Riemannian medians according to the two RD. These distance measures are then applied to the detection of narrow-band sonar signals. Exploration of the translation of measure reference as well as the application of optimum weighting show substantial improvement in detection performance over classical detection method.

This paper presents a unified controller design method for $H^2$ optimal control and model matching problems of linear port-Hamiltonian systems. The controller design problems are formulated as optimization problems on the product manifold of the set of skew symmetric matrices, the manifold of the symmetric positive definite matrices, and Euclidean space. A Riemannian metric is chosen for the manifold in such a manner that the manifold is geodesically complete, i.e., the domain of the exponential map is the whole tangent space for every point on the manifold. In order to solve these problems, the Riemannian gradients of the objective functions are derived, and these gradients are used to develop a Riemannian steepest descent method on the product manifold. The geodesic completeness of the manifold guarantees that all points generated by the steepest descent method are on the manifold. Numerical experiments illustrate that our method is able to solve the two specified problems.

In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

Manifold assumption has been used in example-based super-resolution (SR) reconstruction from a single frame. Previous manifold-based SR approaches (more generally example-based SR) mainly focus on analyzing the co-occurrence properties of low-resolution and high-resolution patches. This paper develops a novel single-image SR approach based on linear approximation of the high-resolution-patch space using a sparse subspace clustering algorithm. The approach exploits the underlying high-resolution patches nonlinear space by considering it as a low-dimensional manifold in a high-dimensional Euclidean space, and by considering each training high-resolution-patch as a sample from the manifold. We utilize the sparse subspace clustering algorithm to create the set of low-dimensional linear spaces that are considered, approximately, as tangent spaces at the high-resolution samples. Furthermore, we consider and analyze each tangent space as one point in a Grassmann manifold, which helps to compute geodesic pairwise distances among these tangent spaces. An optimal subset of these tangent spaces is then selected using a min-max algorithm. The optimal subset reduces the computational cost in comparison with using the full set of tangent spaces while still preserving the quality of the high-resolution image reconstruction. In addition, we perform hierarchical clustering on the optimal subset based on the geodesic distance, which helps to further achieve much faster SR algorithm. We also analytically prove the validity of the geodesic distance based clustering under the proposed framework. A comparison of the obtained results with other related methods in both high-resolution image quality and computational complexity clearly indicates the viability of the proposed framework.

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold $$\overline M $$ of non-negative (n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L q -energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained.

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.

This paper is devoted to a study of geodesic mappings and infinitesimal geodesic deformations of generalized Riemannian spaces. While a geodesic mapping between two generalized Riemannian spaces any geodesic line of one space sends to a geodesic line of the other space, under an infinitesimal geodesic deformation any geodesic line is mapped to a curve approximating a geodesic with a given precision. Basic equations of the theory of geodesic mappings in the case of generalized Riemannian spaces are obtained in this paper. A new generalization of the famous Levi Civita's equation is found. Necessary and sufficient conditions for a nontrivial infinitesimal geodesic deformation are given. It is proven that a generalized Riemannian space admits nontrivial infinitesimal geodesic deformations if and only if it admits nontrivial geodesic mappings. At last it is shown that generalized equidistant spaces of primary type admit nontrivial geodesic deformations.

We prove a normal form result for the groupoid of germs generated by PSL(2,C) and the exponential map. As consequences, we generalize a result of Cohen about the group of translations and powers, and prove that the subgroup of Homeo(R,+infinity) generated by the positive affine maps and the exponential map is isomorphic to a HNN-extension., 47 pages, revised and corrected version accepted for publication

We give a sufficient condition for the essential self-adjointness of a perturbed biharmonic-type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions.

The results of this paper can be viewed as giving a sort of heuristic explanation of why it is so hard to give examples of non-totally geodesic, complete, spacelike, constant mean curvature hypersurfaces M n of a Lorentzian group G n + 1 . More precisely, let N be a timelike unit vector field on M and suppose that the Ricci curvature of G in the direction of N is greater than or equal to − H 2 n , where H is the mean curvature of M with respect to N. If M is compact and transversal to a timelike element of the Lie algebra of G, then we show that it is a lateral class of a Lie subgroup of G and, as such, totally geodesic in G. If M is noncompact and parabolic, then we get the same result, provided M has bounded hyperbolic Gauss map. We also discuss some related examples and, along the way, give a simple proof of the parabolicity of a Riemannian product of a compact and a parabolic Riemannian manifold.

This paper considers optimization problems on Riemannian manifolds and analyzes the iteration-complexity for gradient and subgradient methods on manifolds with nonnegative curvatures. By using tools from Riemannian convex analysis and directly exploring the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, thereby complementing and improving related results. Moreover, we also establish an iteration-complexity bound for the proximal point method on Hadamard manifolds.

The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Rieman- nain manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete non-compact man- ifolds, and, we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.

A finite deformation hyper-elastic membrane theory based on inter-atomic potentials for crystalline films composed of a single atomic layer is developed. For this purpose, an extension of the standard Born rule that exploits the differential geometry concept of the exponential map is proposed to deal with the curvature of surfaces. The exponential map is approximated locally and strain measures based on the stretch and the curvature of the membrane arise. The methodology is first particularized to atomic chains in two dimensions, and then to graphene sheets. A reduced model for the transverse mechanics of carbon nanotubes is developed in detail. This model is a hyper-elastic constrained membrane which fully exploits the symmetry of the transverse deformation. Additionally, a continuum version of the non-bonded interactions is provided. The continuum model is discretized using finite elements and very good agreement with molecular mechanics simulations is obtained. Finally, several simulations illustrate the strong effect of the van der Waals interactions in the transverse deformation of carbon nanotubes.

The Julia set of the exponential family $E_{\kappa}:z\mapsto\kappa e^z$, $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0, Comment: 33 pages, 3 figures, final version, to appear in Ergodic theory and Dynamical Systems

Let $G$ be a complex algebraic group defined over $\mathbb R$, which is not necessarily Zariski connected. In this article, we study the density of the images of the power maps $g\to g^k$, $k\in\mathbb N$, on real points of $G$, i.e., $G(\mathbb R)$ equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of $G(\mathbb R)$. We also characterize surjectivity of the power map for a disconnected group $G(\mathbb R)$. The results are applied in particular to describe the image of the exponential map of $G(\mathbb R).$, Comment: Final version to appear in J. Group Theory

In this work we study the generalization of Stochastic Derivative-Free Optimization (SDFO) algorithms from Euclidean spaces to Riemannian manifolds. In the literature, Riemannian adaptations of SDFO relies on the Riemannian exponential map, which imposes local restrictions. We aim to address this restriction using only the intrinsic geometry of the Riemannian manifold. We first propose Riemannian SDFO (RSDFO), a generalized framework for adapting SDFO algorithms on Euclidean spaces to Riemannian manifolds. We then propose a novel algorithm --- Extended RSDFO, and discuss its convergence behaviour on finite volume Riemann manifolds.

The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model., 31 pages, 1 figure, version appeared in JMP

We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.

Let f1, f2 : CN −→ CN be affine maps fi(x) := Aix + yi (where each Ai is an N -by-N matrix and yi ∈ CN ), and let x1,x2 ∈ AN (C) such that xi is not preperiodic under the action of fi for i = 1, 2. If none of the eigenvalues of the matrices Ai is a root of unity, then we prove that the set {(n1, n2) ∈ N0 : f n1 1 (x1) = f n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. Using this result, we prove that for any two self-maps Φi(x) := Φi,0(x) + yi on a semiabelian variety X defined over C (where Φi,0 ∈ End(X) and yi ∈ X(C)), if none of the eigenvalues of the induced linear action DΦi,0 on the tangent space at 0 ∈ X is a root of unity (for i = 1, 2), then for any two non-preperiodic points x1, x2, the set {(n1, n2) ∈ N0 : Φ n1 1 (x1) = Φ n2 2 (x2)} is a finite union of sets of the form {(m1k + `1,m2k + `2) : k ∈ N0} where m1,m2, `1, `2 ∈ N0. We give examples to show that the above condition on eigenvalues is necessary and introduce certain geometric properties that imply such a condition. Our method involves an analysis of certain systems of polynomial-exponential equations and the padic exponential map for semiabelian varieties.

We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is ℝd or compact without boundary. The main result is that for dim M/2 + 1, the group Ds (M) is geodesically and metrically complete with a surjective exponential map. We also extend the result to its closed subgroups, in particular the group of volume preserving diffeomorphisms and the group of symplectomorphisms. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.

In this paper, an optimization method for nonsmooth locally Lipschitz functions on complete Riemannian manifolds is presented. The method is based on approximating the subdifferential of the cost function at every iteration by the convex hull of transported gradients from tangent spaces at randomly generated nearby points to the tangent space of the current iterate and can hence be seen as a generalization of the well known gradient sampling algorithm to a Riemannian setting. A convergence result is obtained under the assumption that the cost function is bounded below and continuously differentiable on an open set of full measure and that the employed vector transport and retraction satisfy certain conditions, which hold, for instance, for the exponential map and parallel transport. Then with probability one the algorithm produces iterates at which the cost function is differentiable, and each cluster point of the iterates is a Clarke stationary point. Modifications yielding only $\varepsilon$-stationary ...

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 fundamental equations of geodesic mappings of manifolds with affine connection onto (pseudo-)Riemannian manifolds. We proved that if a manifold with affine (or projective) connection of differentiability class $C^r (r\geq2)$ admits a geodesic mapping onto a (pseudo-)Riemannian manifold of class $C^1$, then this manifold belongs to the differentiability class $C^{r+1}$. From this result follows if an Einstein spaces admits non-trivial geodesic mappings onto (pseudo-)Riemannian manifolds of class $C^1$ then this manifold is an Einstein space, and there exists a common coordinate system in which the components of the metric of these Einstein manifolds are real analytic functions.

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.

We study some basic properties related to affine functions on Riemannian manifolds. A characterization for a function to be linear affine is given and a counterexample on Poincare plane is provided, which, in particular, shows that assertions (i) and (ii) claimed by Papa Quiroz in (J Convex Anal 16(1):49---69, 2009, Proposition 3.4) are not true, and that the function involved in assertion (ii) is indeed not quasi-convex. Furthermore, we discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures.

In this paper, we discuss the optimal H2 model order reduction (MOR) problem for bilinear systems. The H2 optimal MOR problem of bilinear systems is considered as the minimisation problem on Grassmann manifold, which is stored as a quotient space of the noncompact Stifiel manifold. Grassmann manifold whose tangent space is endowed with a Riemannian metric is a Riemannian manifold. In its tangent space equipped with the Riemannian metric, we derive the negative gradients of the cost function, i.e. the steepest descent direction of the cost function. After that, the formulas of geodesic on Grassmann manifold are given. Then we perform linear searches along geodesics to obtain the optimal solutions. Thereby, a two-sided MOR iterative algorithm is proposed to construct an order-reduced bilinear system, which is used to simulate the output and input responses of the original bilinear system. Numerical examples demonstrate the effectiveness of our algorithm.