Real analytic metrics on S 2 with total absence of finite blocking

If (M, g) is a Riemannian manifold and ${(x,y) \in M \times M,}$ then a set ${P \subset M \backslash \{x,y\}}$ is said to be a blocking set for (x, y) if every geodesic from x to y passes through a point of P. If no pair (x, y) in M × M has a finite blocking set, then (M, g) is said to be totally insecure. We prove that there exist real analytic metrics h on S ² such that (S ², h) is totally insecure. Previously known examples of totally insecure metrics on S ² were only C ¹.     

Riemannian and Sub-Riemannian Geodesic Flows

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 result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

A note on the divergence of geodesic rays in manifolds without conjugate points

Let (M, g) be a compact Riemannian manifold without conjugate points and let be its universal covering endowed with the pullback of the metric g by the covering map. We show that geodesic rays in which meet an axis of a covering isometry diverge from this axis. This result generalizes well known results by Morse and Hedlund in the context of globally minimizing geodesics in the universal covering of compact surfaces.      

A Support Theorem for the Geodesic Ray Transform on Functions

Let (M,g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f∈L ²(M) vanishes on an appropriate open set of geodesics, then f=0 on the set of points lying on these geodesics. The approach is based on analytic microlocal analysis.     

Entropy of Semiclassical Measures for Nonpositively Curved Surfaces

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov–Sinai entropy of a semiclassical measure μ for the geodesic flow g ^t is bounded from below by half of the Ruelle upper bound, i.e.

The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> topology

We show that the set of C metrics in the two dimensional torus with no continuous invariant graphs of the geodesic flow is open and dense in the C ¹ topology. The generic nonexistence of invariant graphs with rational rotation numbers was known in the C topology for metrics, and in general the generic nonexistence in the C topology of invariant graphs with Liouville rotation numbers is known for twist maps and Hamiltonian flows in the torus. The main idea of the proof is that small C ¹ bumps are enough to prevent the existence of invariant graphs.Partially supported by CNPq, FAPERJ, TWAS     

Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle M of complex spinormodules over the Clifford bundle Cl(g) of a Riemannian space(M, g) with local model (V, h)is associated with an lpin(Lipschitz) structure on M, this being a reduction of theO(h)-bundle of all orthonormal frames on M to the Lipschitzgroup Lpin(h) of all automorphisms of a suitably defined spinspace. An explicit construction is given of the total space of theLpin(h)-bundle defining such a structure. If the dimension mof M is even, then the Lipschitz group coincides with the complexClifford group and the lpin structure can be reduced to a pin _c structure. If m = 2n – 1, then a spinor module on M is of the Cartan type: its fibres are 2 _n -dimensional anddecomposable at every point of M, but the homomorphism of bundlesof algebras Cl(g) End globally decomposes if, andonly if, M is orientable. Examples of such bundles are given. Thetopological condition for the existence of an lpin structure on anodd-dimensional Riemannian manifold is derived and illustrated by theexample of a manifold admitting such a structure, but no pin _c structure.     

Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L²(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L2([0,T]×Ω), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of Ω. We prove C?d²/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove for some . Moreover, this bound implies where is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest.     

Classes of submersions of Riemannian manifolds with compact fibers

In Riemannian geometry and its applications, the most popular is the class of Riemannian submersions (and foliations) [1–4] which are characterized by simplest mutual disposition of fibers. The purpose of the present article is to introduce other, more general, classes of submersions of Riemannian manifolds which, as well as the class of Riemannian submersions, are described by simple local properties of configuration tensors and to begin their study.Given a submersion :MM of differentiable manifolds with compact connected fibers and any metric onM, we define a metric on the base with the help of theL ²-norm of horizontal fields. In this caseT¯ M becomes a subbundle of some larger bundleM. The main class of totally geodesic submersions introduced in the article (Definition 1) corresponds to the metrics onM with simplest disposition ofT¯ M inM. In the article we obtain a criterion for such submersions (Corollary 1); existence is proved by means of the product with a metric varying along fibers (Example 2). To study totally geodesic submersions, we use ideas from the theory of Riemannian submersions and submanifolds with degenerate second form (Theorems 1 and 2 and Corollary 4).Foliations modeled by totally geodesic submersions (see equality (13)) are of interest too, but we leave them beyond the scope of the article.This work was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00271).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1154–1164, September–October, 1994.     

Notes on Gromov’s systolic estimate

These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C _n Vol(M,g)^1/n .     

On the Relation between Curvature,Diameter, and Volume of a Complete Riemannian Manifold

In this note, we prove that if N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then for any point m ∈ M. In the case where dim M = 2, the Gaussian curvature K satisfies the relation K ≥ k ≥ 0, and γ is of length l, we get Vol (M, g) ≤ if k ≠ 0 and Vol (M, g ≤ 2ldiam (M) if k = 0.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1576–1583, November, 2004.     

The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

On Riemannian g-natural Metrics of the Form a.g s + b.g h + c.g v on the Tangent Bundle of a Riemannian Manifold (M, g)

Let (M, g) be a Riemannian manifold and TM its tangent bundle. In [5] we have investigated the family of all Riemannian g-natural metrics G on TM (which depends on 6 arbitrary functions of the norm of a vector u TM). In this paper, we continue this study under some additional geometric properties, and then we restrict ourselves to the subfamily {G=a.g^s + b.g^h + c.g^v, a, b and c are constants satisfying a > 0 and a(a + c) – b² > 0}. It is known that the Sasaki metric g^s is extremely rigid in the following sense: if (TM, g^s) is a space of constant scalar curvature, then (M, g) is flat. Here we prove, among others, that every Riemannian g-natural metric from the subfamily above is as rigid as the Sasaki metric.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result

Let (M,g) be an n-dimensional (n?2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C^∞. This paper is concerned with the study of the wave equation on (M,g) with locally distributed damping, described by

     

On Ricci curvature ofC-totally real submanifolds in Sasakian space forms

LetM ⁿ be a Riemanniann-manifold. Denote byS(p) and Ric(p) the Ricci tensor and the maximum Ricci curvature onM _n, respectively. In this paper we prove that everyC-totally real submanifold of a Sasakian space formM ^2m+1(c) satisfies , whereH ² andg are the square mean curvature function and metric tensor onM ⁿ, respectively. The equality holds identically if and only if eitherM ⁿ is totally geodesic submanifold or n = 2 andM ⁿ is totally umbilical submanifold. Also we show that if aC-totally real submanifoldM ⁿ ofM ²ⁿ⁺¹ (c) satisfies identically, then it is minimal.     

Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if ${H \subset M}$ is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions ${\varphi_j |_H}$ of an orthonormal basis ${\{\varphi_j\}}$ of Δ-eigenfunctions of (M, g) to H are quantum ergodic on H. The condition on H is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface. A key step in the proof is that matrix elements ${\langle{F}\varphi_j, \varphi_{j}\rangle}$ of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.     

On 3-dimensional asymptotically harmonic manifolds

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature , provided M is asymptotically harmonic of constant h > 0. Received: 4 October 2007