Manifold Of Nonpositive Sectional Curvature Research Articles

ON SOME MANIFOLDS OF CONSTANT NEGATIVE CURVATURE

Manifold of constant negative curvature a great role in the field of physics, mathematics and engineering because it paves to the knowledge Gaussian curvature , n-sphere(s) is a topological n-manifold and objects of constant negative curvature are less familiar but they do appear in nature in the shape of corals and leave . No surprisingly , its plays an important role in geometric topology. The study aims to explain a generalization manifold of constant negative curvature . WE followed the analytical induction mathematical method . We found the following some results. Manifold of constant negative curvature indicates to know the behavior of some of the functions and also it reveals the Cartan-Hadamard theorem which is considered one of the importance aims of simply-connected manifold of nonpositive sectional curvature.

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds.

Price inequalities and Betti number growth on manifolds without conjugate points

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.

The coarse geometric $\ell^p$-Novikov conjecture for subspaces of nonpositively curved manifolds

In this paper, we prove the coarse geometric \ell^p -Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions

We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the second author [18], under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi [12] by a factor of $(\log \lambda)^{\mu}$ for any $\mu \lt \frac{2(n+1)^2}{n-1}$, if $n \geq 3$.

Translation Lengths of Parabolic Isometries of CAT(0) Spaces and Their Applications

In this article, we provide a sufficient and necessary condition on parabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

A theorem of Liouville type for p-harmonic maps in weighted Riemannian manifolds

Let M be a weighted Riemannian manifold with non-negative Bakry-Emery-Ricci curvature and N be a complete Riemannian manifold of non-positive sectional curvature. In this paper, the p-harmonic map u: M → N is studied, and a theorem of Liouville type is obtained.

Half-flat structures inducing Einstein metrics on homogeneous spaces

In this paper, we consider half-flat \(\mathrm{SU}(3)\)-structures and the subclasses of coupled and double structures. In the general case, we show that the intrinsic torsion form \(w_1^-\) is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left-invariant half-flat (non-coupled and non-double) structure inducing an Einstein metric on \(S^3\times S^3\) and we show there does not exist any left-invariant coupled structure inducing an \(\mathrm{Ad}(S^1)\)-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.

Fibred coarse embedding into non-positively curved manifolds and higher index problem

Let X=⨆n=1∞Xn be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. In this paper, we show that the coarse Novikov conjecture holds for X, if X admits a fibred coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature. This includes a large class of expander graphs with geometric property (T).

On the Dirichlet problem for $$p$$ p -harmonic maps I: compact targets

In this paper we solve the relative homotopy Dirichlet problem for $$p$$ -harmonic maps from compact manifolds with boundary to compact manifolds of non-positive sectional curvature. The proof, which is based on the direct calculus of variations, uses some ideas of B. White to define the relative d-homotopy type of Sobolev maps. One of the main points of the proof consists in showing that the regularity theory by Hardt and Lin can be applied. A comprehensive uniqueness result for general complete targets with non-positive curvature is also given.

LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

Abstract. Let M be a complete Riemannian manifold and let N be aRiemannian manifold of non-positive sectional curvature. Assume thatRic M ≥ − 4(p−1)p 2 µ 0 at all x ∈ M and Vol(M) is infinite, where µ 0 > 0 isthe infimum of the spectrum of the Laplacian acting on L 2 -functions onM. Then any p-harmonic map φ : M → N of finite p-energy is constant.Also, we study Liouville type theorem for p-harmonic morphism. 1. IntroductionLet (M,g) and (N,h) be smooth Riemannian manifolds and let φ : M → Nbe a smooth map. For a compact domain Ω ⊂ M, the p-energy E p (φ;Ω) of φover Ω is defined by(1.1) E p (φ;Ω) =1pZ Ω |dφ| p µ M ,where the differential dφ is a section of the bundle T ∗ M ⊗ φ −1 TN → M andφ −1 TN denotes the pull-back bundle via the map φ. The bundle T ∗ M ⊗φ −1 TN → M carries the connection ∇ induced by the Levi-Civita connectionson M and N. A map φ : M → N is called p-harmonic if the p-tension fieldτ p (φ) = 0, which is defined by(1.2) τ p (φ) = tr g ∇(|dφ| p−2 dφ),where tr g denote the trace with respect to the metric g. A p-harmonic map φ isa critical point of the energy functional defined by (1.1) on any compact domainΩ ⊂ M. When p = 2, p-harmonic maps are well-known to be harmonic maps.Several studies are given for harmonic maps (see [5], [6], [7], [8], [10], [11], [12],[13], [14], [16]). Let µ

Hofer geometry and cotangent fibers

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positivecurvature examples such as spheres of dimension at least 3 and compact semisimple Lie groups on the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle contains as subgroups infinite-dimensional normed vector spaces that are bi-Lipschitz embedded with respect to Hofer’s metric; moreover these subgroups can be taken to consist of diffeomorphisms supported in an arbitrary neighborhood of the zero section. In fact, the orbit of a fiber of the cotangent bundle with respect to any of these subgroups is quasi-isometrically embedded with respect to the induced Hofer metric on the orbit of the fiber under the whole group. The diffeomorphisms in these subgroups are obtained from reparametrizations of the geodesic flow. Our proofs involve a study of the Hamiltonian-perturbed Floer complex of a pair of cotangent fibers (or, more generally, of a conormal bundle together with a cotangent fiber). Although the homology of this complex vanishes, an analysis of its boundary depth yields the lower bounds on the Lagrangian Hofer metric required for our main results.

4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings

We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature.

An invariant of nonpositively curved contact manifolds

Abstract We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can be approximated by isotopies of a given contact structure.

Total curvature of curves in Riemannian manifolds

The total curvature of C 2 curves embedded in an arbitrary Riemannian manifold is shown to be the limit of the curvatures of inscribed geodesic polygons. The formula for the total curvature of a curve as the least upper bound of curvatures of inscribed geodesic polygons holds for a manifold of non-positive sectional curvature only.

Monotone vector fields and the proximal point algorithm on Hadamard manifolds

The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to solve that problem, which extends the known proximal point algorithm in Euclidean spaces. We prove that the sequence generated by our method is well defined and converges to a singularity of a maximal monotone vector field, whenever it exists. Applications in minimization problems with constraints, minimax problems and variational inequality problems, within the framework of Hadamard manifolds, are presented.

Fundamental classes not representable by products

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.

On the behavior of a simple-layer potential for a parabolic equation on a Riemannian manifold

On a Riemannian manifold of nonpositive sectional curvature (Cartan-Hadamard-type manifold), we consider a parabolic equation. The second boundary-value problem for this equation is set in a bounded domain whose surface is a smooth submanifold. We prove that the gradient of the simple-layer potential for this problem has a jump when passing across the submanifold, similarly to its behavior in a Euclidean space.

HORIZONTALLY HOMOTHETIC HARMONIC MORPHISMS AND STABILITY OF TOTALLY GEODESIC SUBMANIFOLDS

In this article, we study the relations of horizontally homothetic harmonic morphisms with the stability of totally geodesic submanifolds. Let φ : (Mn, g) → (Nm, h) be a horizontally homothetic harmonic morphism from a Riemannian manifold into a Riemannian manifold of non-positive sectional curvature and let T be the tensor measuring minimality or totally geodesics of fibers of φ. We prove that if T is parallel and the horizontal distribution is integrable, then for any totally geodesic submanifold P in N , the inverse set, φ−1(P ), is volume-stable in M . In case that P is a totally geodesic hypersurface, the condition on the curvature can be weakened to Ricci curvature.

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that Ric M ⩾ − 4 ( p − 1 ) p 2 μ 0 at all x ∈ M and > − 4 ( p − 1 ) p 2 μ 0 at some point x 0 ∈ M , where μ 0 > 0 is the least eigenvalue of the Laplacian acting on L 2 -functions on M. Let 2 ⩽ q ⩽ p . Then any q-harmonic map ϕ : M → N of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism ϕ : M → N of finite q-energy is constant.