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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.

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

In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set of escaping endpoints joined with infinity is connected for any transcendental entire function of finite order with bounded singular set. We also discuss corresponding results for *all* endpoints in the case of exponential maps; in order to do so, we establish a version of Thurston's "no wandering triangles" theorem., Comment: 35 pages. To appear in Comput. Methods Funct. Theory. V2: Authors' final accepted manuscript. Revisions and clarifications have been made throughout from V1. This includes improvements in the proof of Proposition 6.11 and Theorem 8.1, as well as corrections in Remarks 7.1 and 7.3 concerning differing definitions of escaping endpoints in greater generality

We demonstrate that the surface quasi-geostrophic (SQG) equation given by $$\theta_t + \left= 0,\;\;\; \theta = \nabla \times (-\Delta)^{-1/2} u,$$ is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold $M$ in the right-invariant $\dot{H}^{-1/2}$ metric. We show by example, that the Riemannian exponential map is smooth and non-Fredholm, and that the sectional curvature at the identity is unbounded of both signs.

In the present paper we study equitorsion geodesic mappings between two generalized Riemannian spaces f : GR N → G R ‾ N . In this case these spaces have the same torsions at corresponding points. We prove that a generalized Riemannian space GR N admits an equitorsion geodesic mapping onto a generalized Riemannian space G R ‾ N if and only if in GR N certain differential equations of Cauchy type in covariant derivatives of the θ = 1 , … , 4 kinds have a solution with respect to the unknown tensor a i j , the gradient vector λ i ≠ 0 , and the differentiable function μ θ , θ = 1 , … , 4 . In fact we find four systems of PDE all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of PDE. We establish an upper bound for the number of solutions for the geometrical problem under consideration.

In this paper, we have studied harmonic maps on trans-Sasakian manifolds. First it is proved that if F: M1 → M2 is a Riemannian ϕ-holomorphic map between two trans-Sasakian manifolds such that ξ2 ∈ (Im dF)⊥, then F can not be harmonic provided that β2 ≠ 0. We have also found the necessary and sufficient condition for the harmonic map to be constant map from Kaehler to trans-Sasakian manifold. Finally, we prove the non-existence of harmonic map from locally conformal Kaehler manifold to trans-Sasakian manifold.

In this paper, we derive a local Hamilton type gradient estimate for a nonlinear heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem.

Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.

In this Note, we study an optimal control problem on a Riemannian manifold. The control set in our problem is assumed to be a general Polish space, and therefore the classical variation technique fails. We obtain a pointwise second-order optimality condition, for which the curvature tensor of the manifold appears explicitly in the second-order dual equation.

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 .

We provide a direct proof of Cram\'er's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into and overcome these by providing Taylor expansions of the inverse Riemannian exponential map, while also comparing the differential of the exponential map to parallel transport. Finally, we obtain the analogue of Cram\'er's theorem for geodesic random walks by showing that the curvature terms arising in this geometric analysis can be controlled and are negligible on an exponential scale., Comment: A mistake in the proof has been corrected. To appear in Electronic Journal of Probability

This papers aims at describing a novel framework for bi-invariant density estimation on the group of planar rigid motion SE(2). Probability distributions on the group are constructed from distributions on tangent spaces pushed to the group by the exponential map. The exponential mapping on Lie groups presents two key particularities: it is compatible with left and right multiplications and its Jacobian can be computed explicitly. These two properties enable to define probability densities with tractable expressions and bi-invariant procedures to estimate them from a set of samples. Sampling from these distributions is easy since it is sufficient to draw samples in Euclidean tangent spaces. This paper is a preliminary work and the convergences of these estimators are not studied.

The monograph deals with the theory of conformal, geodesic, holomorphically projective, F-planar and others mappings and transformations of manifolds with affine connection, Riemannian, Kahler and Riemann-Finsler manifolds. 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.

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics., Comment: 7 pages. Comments welcome

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold 𝔐 {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of 𝔐 {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in ℝ N {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

This article develops variational integrators for a class of underactuated mechanical systems using the theory of discrete mechanics. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. Exploiting the left-trivialization of the cotangent bundle, and assuming the time-step of discrete evolution is small enough to exploit the diffeomorphism feature of the exponential map in a neighbourhood of the identity of the Lie group, that enables a mapping of the group variables to the Lie algebra, a variational approach is adopted to obtain the first order necessary conditions that characterise optimal trajectories. The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments.

This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and also a research paper on bi-harmonic maps principal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifold , every pseudo bi-harmonic isometric immersion into a Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature.

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.

A divergence function on a manifold M defines a Riemannian metric g and dually coupled affine connections ∇ and ∇ * on M. When M is dually flat, that is flat with respect to ∇ and ∇ * , a canonical divergence is known, which is uniquely determined from ( M , g , ∇ , ∇ * ) . We propose a natural definition of a canonical divergence for a general, not necessarily flat, M by using the geodesic integration of the inverse exponential map. The new definition of a canonical divergence reduces to the known canonical divergence in the case of dual flatness. Finally, we show that the integrability of the inverse exponential map implies the geodesic projection property.

Here we investigate some geometric properties of the contactomorphism group \(\mathcal {D}_\theta (M)\) of a compact contact manifold with the \(L^2\) metric on the stream functions. Viewing this group as a generalization to the \(\mathcal {D}(S^1)\), the diffeomorphism group of the circle, we show that its sectional curvature is always non-negative and that the Riemannian exponential map is not locally \(C^1\). Lastly, we show that the quantomorphism group is a totally geodesic submanifold of \(\mathcal {D}_\theta (M)\) and talk about its Riemannian submersion onto the symplectomorphism group of the Boothby-Wang quotient of M.

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.

In this note we show that the moduli space M(Sg,n) of surface Sg,n of genus g with n punctures, satisfying 3g + n ≥ 5, admits no complete Riemannian metric of nonpositive sectional curvature such that the Teichmuler space T(Sg,n) is a mapping class group Mod(Sg,n)-invariant visibility manifold.

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

We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by Garcia-Rio and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Aahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O'Neill's book and Falcitelli, Ianus and Pastore's book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu's book to obtain some characterizations and to give a semi-Riemannian version of Fischer's (resp. Aahin's) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.

We derive and apply a new integral formula for a closed Riemannian manifold equipped with a pair of complementary orthogonal distributions (plane fields). The integrand depends on the second fundamental forms and integrability tensors of the distributions, their covariant derivatives, and of the Ricci curvature of the ambient manifold. Also, we discuss some applications of this formula and of another formula of this sort, the one obtained earlier by the second author, and show that both formulae may hold when the distributions are defined only outside a “reasonable” closed subset of the manifold under consideration.

In the present paper we study the class of harmonic maps on cosymplectic manifolds. First we find the necessary and sufficient condition for the Riemannian map to be harmonic map between two cosymplectic manifolds and then from cosymplectic manifold to Sasakian manifold. Finally, we find the condition for non-existence of harmonic map from cosymplectic manifold to Kenmotsu manifold.

Let \(\mathcal {D}_{\omega }^{s}(M)\) denote the group of symplectic diffeomorphisms of a closed symplectic manifold \(M\), which are of Sobolev class \(H^{s}\) for sufficiently high \(s\). When equipped with the \(L^{2}\) metric on vector fields, \(\mathcal {D}_{\omega }^{s}\) becomes an infinite-dimensional Hilbert manifold whose tangent space at a point \(\eta \) consists of \(H^{s}\) sections \(X\) of the pull-back bundle \(\eta ^{*}TM\) for which the corresponding vector field \(u=X\circ \eta ^{-1}\) on \(M\) satisfies \(\mathcal {L}_{u}\omega =0\). Geodesics of the \(L^{2}\) metric are globally defined, so that the \(L^{2}\) metric admits an exponential mapping defined on the whole tangent space. It was shown that this exponential mapping is a non-linear Fredholm map of index zero. Singularities of the exponential map are known as conjugate points and in this paper we construct explicit examples of them on \(\mathcal {D}_{\omega }^{s}(\mathbb {C}P^{n})\). We then solve the Jacobi equation explicitly along a geodesic in \(\mathcal {D}_{\omega }^{s}\), generated by a Killing vector field, and characterize all conjugate points along such a geodesic.

The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.

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.

In this paper we seek to exploit information geometry in order to define the Riemannian metric of the manifold associated with nonparametric active contour models from the exponential family. This Riemannian metric is obtained through a relationship between the contour's energy functional and the likelihood of the categorical latent variables of a mixture model. Accordingly contours form a statistical manifold equipped with a natural metric which is determined by the model's Fisher information matrix. Mathematical developments show that this matrix has a closed-form analytic expression and is diagonal. Based on this, we subsequently develop a Riemannian steepest descent algorithm for the active contour, with application to image segmentation. Because the proposed method performs optimisation on the parameter's natural manifold it attains dramatically faster convergence rates than the Euclidean gradient descent algorithm commonly used in the literature. A segmentation experiment on an ultrasound image is presented and confirms that the proposed natural gradient algorithm converges extremely fast and delivers accurate segmentation results in few iterations.

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).

For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian (1 < p < +∞) on M. This result can be seen as an extension of Reilly’s bound for the first non-zero closed eigenvalue of the Laplace operator.

An approach for clustering multivariate time series (MTS) is presented in cases of variable length, noisy data or mix of different type variables. First the covariance matrices are estimated which is used as a feature to represent the MTS, then project the covariance matrices from a Riemannian manifold into a tangent space and finally carry out the clustering based on a distance matrix. In this procedure, a geodesic-based distance is also introduced for measuring the similarity between the MTS samples. The proposed approach on a chaotic MTS with known clustering structure, namely Lorenz system is evaluated.

Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the $K$-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator., Comment: added details in proofs, text as in Commun. Math. Phys

In this paper, we propose a novel construction of Bezier curves of two-dimensional (\(2\)D) orientations using the geometry of real projective plane \(\mathrm{\mathbb {R}P^{2}}\). Unlike the commonly adopted unit 2-sphere model \(S^{2}\), \(\mathrm{\mathbb {R}P^{2}}\) is naturally embedded in the \(3\)D special orthogonal group \(\mathrm{SO(3)}\). It is also a symmetric space that is equipped with a particular class of isometries called geodesic symmetry, which allows us to generate any geodesics using the exponential map of \(\mathrm{SO(3)}\). We implement the generated geodesics to construct Bezier curves for direction interpolation.

On each Riemannian or pseudo-Riemannian manifold, there is a unique connection determined by the metric, called the Levi-Civita connection. After defining it, we investigate the exponential map, which conveniently encodes the collective behavior of geodesics and allows us to study the way they change as the initial point and initial velocity vary.

Adaptive regularization with cubics (ARC) is an algorithm for unconstrained, non-convex optimization. Akin to the popular trust-region method, its iterations can be thought of as approximate, safe-guarded Newton steps. For cost functions with Lipschitz continuous Hessian, ARC has optimal iteration complexity, in the sense that it produces an iterate with gradient smaller than $\varepsilon$ in $O(1/\varepsilon^{1.5})$ iterations. For the same price, it can also guarantee a Hessian with smallest eigenvalue larger than $-\varepsilon^{1/2}$. In this paper, we study a generalization of ARC to optimization on Riemannian manifolds. In particular, we generalize the iteration complexity results to this richer framework. Our central contribution lies in the identification of appropriate manifold-specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction. A substantial part of the paper is devoted to studying these assumptions---relevant beyond ARC---and providing user-friendly sufficient conditions for them. Numerical experiments are encouraging., Comment: 48 pages, 3 figures

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle TX of a complex manifold X makes the shifted tangent bundle TX[−1] into a Lie algebra object in the derived category D(X). Moreover, he showed that there is an L∞-algebra structure on the shifted Dolbeault resolution (A •−1 X (TX), ∂) of TX and wrote down the structure maps explicitly in the case when X is Kahler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution (A X (J ∞ X ), ∂) of the jet bundle J ∞ X via the construction of the holomorphic exponential map of the Kahler manifold. In this paper, we show that (A X (J ∞ X ), ∂) is naturally isomorphic to the Dolbeault dga (A•(X X×X), ∂) associated to the formal neighborhood of the diagonal of X× X which we introduced in [Yu12]. We also give an alternative proof of Kapranov’s theorem by obtaining an explicit formula for the pullback of functions via the holomorphic exponential map, which allows us to study the general case of an arbitrary embedding later.

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.

The present paper elaborates on tangent-bundle maps on the Grassmann manifold, with application to subspace arithmetic averaging. In particular, the present contribution elaborates on the work about retraction/lifting maps devised for the Stiefel manifold in the recently published paper T. Kaneko, S. Fiori and T. Tanaka, “Empirical arithmetic averaging over the compact Stiefel manifold,” IEEE Trans. Signal Process., Vol. 61, No. 4, pp. 883–894, February 2013, and discusses the extension of such maps to the Grassmann manifold. Tangent-bundle maps are devised on the basis of the thin QR matrix decomposition, the polar matrix decomposition and the exponential map. Also, tangent-bundle pseudo-maps based on the matrix Cayley transform are devised. Theoretical and numerical comparisons about the devised tangent-bundle maps are performed in order to get an insight into their relative merits and demerits, with special emphasis to their computational burden. The averaging algorithm based on the thin-QR decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden. Such algorithm is further compared with two Grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set. The thin-QR tangent-bundle maps-based algorithm exhibits again numerical features that make it preferable over such algorithms.