Riemannian manifolds

We prove that knowing the lengths of geodesics joining points of the boundary of a twodimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction. 1. Introduction and statement of the results Let (M, g) be a compact Riemannian manifold with boundary ∂M . Let dg (x, y) denote the geodesic distance between x and y. The inverse problem we address in this paper is whether we can determine the Riemannian metric g knowing dg (x, y) for any x ∈ ∂M , y ∈ ∂M . ...
19p noel_noel 17012013 20 3 Download

For each k ∈ Z, we construct a uniformly contractible metric on Euclidean space which is not mod k hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on Rn , n ≥ 11, so that the resulting manifolds Z and Z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We show that for these lf spaces the C ∗ algebra assembly map K∗ (Z) → K∗ (C ∗ (Z)) from locally ﬁnite Khomology to the Ktheory of the bounded propagation algebra is not a monomorphism ...
21p tuanloccuoi 04012013 12 5 Download

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article ´ BrezisWainger Inequality on Riemannian Manifolds
6p sting09 22022012 15 2 Download

The Ricci ﬂow was introduced by Hamilton in 1982 [H1] in order to prove that a compact threemanifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact fourmanifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact fourmanifolds with 2positive curvature operators [Che]. Recall that a curvature operator is called 2positive, if the sum of its two smallest eigenvalues is positive. ...
20p dontetvui 17012013 26 7 Download

We deﬁne and study an algebra Ψ∞ (M0 ) of pseudodiﬀerential opera1,0,V tors canonically associated to a noncompact, Riemannian manifold M0 whose geometry at inﬁnity is described by a Lie algebra of vector ﬁelds V on a compactiﬁcation M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodiﬀerential operators on a compact manifold extend to Ψ∞ (M0 ).
32p noel_noel 17012013 23 5 Download

There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diﬀeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold (M, g) is up to homotopy given by a rank one symmetric space, provided that its isometry group Iso(M, g) is large. More precisely we prove ﬁrst that if dim(Iso(M, g)) ≥ 2 dim(M ) − 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M...
63p noel_noel 17012013 25 4 Download

We study enhancement of diﬀusive mixing on a compact Riemannian manifold by a fast incompressible ﬂow. Our main result is a sharp description of the class of ﬂows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the ﬂow amplitude is large enough. The necessary and suﬃcient condition on such ﬂows is expressed naturally in terms of the spectral properties of the dynamical system associated with the ﬂow. In particular, we ﬁnd that weakly mixing ﬂows always enhance dissipation in this sense. ...
33p dontetvui 17012013 26 8 Download

Let X = G/K be a homogeneous Riemannian manifold where G is the identity component of its isometry group. A C ∞ function F on X is harmonic if it is annihilated by every element of DG (X), the algebra of all Ginvariant diﬀerential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture, which states that on a Riemannian symmetric space of noncompact type, a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G/Po where...
59p tuanloccuoi 04012013 23 6 Download

This paper is the ﬁrst in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed Riemannian 3manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the ﬂat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of ﬁxed genus surfaces in 3manifolds. There are two local models for...
43p tuanloccuoi 04012013 33 6 Download

We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature – in fact, we only assume that the geodesic ﬂow has the Anosov property. In the semiclassical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesics. 1.
43p dontetvui 17012013 22 6 Download

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have a nontrivial amount of symmetry. By this we mean that Isom(M ) is not discrete. By the wellknown theorem of MyersSteenrod [MS], this condition is equivalent to [Isom(M ) : π1 (M )] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps,...
27p dontetvui 17012013 30 6 Download

Let T (x, ε) denote the ﬁrst hitting time of the disc of radius ε centered at x for Brownian motion on the two dimensional torus T2 . We prove that supx∈T2 T (x, ε)/ log ε2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, twodimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z2 is asymptotic to 4n2 (log...
33p tuanloccuoi 04012013 19 5 Download

This paper is a continuation of Feﬀerman’s program [7] for studying the geometry and analysis of strictly pseudoconvex domains. The key idea of the program is to consider the Bergman and Szeg¨ kernels of the domains as o analogs of the heat kernel of Riemannian manifolds. In Riemannian (or conformal) geometry, the coeﬃcients of the asymptotic expansion of the heat kernel can be expressed in terms of the curvature of the metric; by integrating the coeﬃcients one obtains index theorems in various settings. ...
18p noel_noel 17012013 21 5 Download

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric. 1. Introduction Let (M n , g) be a smooth, closed Riemannian manifold of dimension n.
58p noel_noel 17012013 30 5 Download

Invariant measures for the geodesic ﬂow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and wellstudied subject. This paper continues an investigation into a 2dimensional analog of this ﬂow for a 3manifold N . Namely, the article discusses 2dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1, surfaces which are called ksurfaces.
37p noel_noel 17012013 20 4 Download

Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior X ◦ the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Rn . Consider the operator H = 1 ∆ + V , where ∆ is the positive Laplacian with respect to g and V is a 2 smooth realvalued function on X vanishing to second order at ∂X. Assuming that g is nontrapping, we construct a global parametrix U(z, w, t) for the kernel...
38p noel_noel 17012013 22 4 Download

Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds: An Introduction to Relativity 8.
128p khangoc2391 11082012 19 1 Download

In this paper we study periodic solutions of the equation \begin{equation}\label{a} \frac{1}{i}\left( \frac{\partial}{\partial t}+aA \right)u(x,t)=\nu G (uf), \end{equation} with conditions \begin{equation}\label{b} u_{t=0}=u_{t=b}, \,\, \int_X (u(x),1) \, dx =0 \end{equation} over a Riemannian manifold $X$, where $$G u(x,t)=\int_Xg(x,y)u(y)dy $$ is an integral operator, $u(x,t)$ is a differential form on $X,$ $A=i(d+\delta)$ is a natural differential operator in $X$. We consider the case when $X$ is a tore $\Pi^2$.
11p tuanlocmuido 19122012 12 1 Download

SAMUEL EILENBERG. Automata, Languages, and Machines: Volumes A and B MORRIS HIRSCH N D STEPHEN A SMALE. Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS. Noneuclidean Tesselations and Their Groups FRANCOIS TREVES. Linear Partial Differential Equations Basic WILLIAM BOOTHBY. Introduction to Differentiable Manifolds and Riemannian M. An Geometry ~ A Y T O N GRAY. Homotopy Theory : An Introduction to Algebraic Topology ROBERT ADAMS. A. Sobolev Spaces JOHN BENEDETTO. J. Spectral Synthesis D. V. WIDDER.
291p mrbin1262006 06052010 86 23 Download

Institute for Theoretical Physics University of California Santa Barbara, CA 93106 carroll@itp.ucsb.edu December 1997 Abstract These notes represent approximately one semester’s worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: gravitational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at http://itp.ucsb.edu/~carroll/notes/. NSFITP/97147 grqc/9712019 .i Table of Contents 0.
238p possibletb 23112012 38 7 Download