# Riemannian manifolds

Xem 1-20 trên 20 kết quả Riemannian manifolds
• ### Đề tài " Two dimensional compact simple Riemannian manifolds are boundary distance rigid "

We prove that knowing the lengths of geodesics joining points of the boundary of a two-dimensional, 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 . ...

• ### Đề tài " Large Riemannian manifolds which are flexible "

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 K-homology to the K-theory of the bounded propagation algebra is not a monomorphism ...

• ### Báo cáo hóa học: "Research Article ´ Brezis-Wainger Inequality on Riemannian Manifolds"

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 ´ Brezis-Wainger Inequality on Riemannian Manifolds

• ### Đề tài " Manifolds with positive curvature operators are space forms "

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

• ### Đề tài "Pseudodifferential operators on manifolds with a Lie structure at infinity "

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 ).

• ### Đề tài " Positively curved manifolds with symmetry "

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...

• ### Đề tài " Di_usion and mixing in uid ow "

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. ...

• ### Đề tài " Van den Ban-SchlichtkrullWallach asymptotic expansions on nonsymmetric domains "

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 G-invariant 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...

• ### Đề tài " The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks "

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 3-manifold. 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 3-manifolds. There are two local models for...

• ### Đề tài " Entropy and the localization of eigenfunctions "

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 semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesics. 1.

• ### Đề tài " Isometries, rigidity and universal covers "

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 well-known theorem of Myers-Steenrod [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,...

• ### Đề tài " Cover times for Brownian motionand random walks in two dimensions "

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 ε|2 → 2/π as ε → 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, 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...

• ### Đề tài " Logarithmic singularity of the Szeg¨o kernel and a global invariant of strictly pseudoconvex domains "

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. ...

• ### Đề tài " Prescribing symmetric functions of the eigenvalues of the Ricci tensor "

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.

• ### Đề tài " Random k-surfaces "

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

• ### Đề tài "The Schr¨odinger propagator for scattering metrics "

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 real-valued 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...

• ### Introduction to Differential Geometry & General Relativity Third Printing January 2002

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.

• ### Báo cáo " Periodic solutions of some linear evolution systems of natural differential equations on 2-dimensional tore "

In this paper we study periodic solutions of the equation $$\label{a} \frac{1}{i}\left( \frac{\partial}{\partial t}+aA \right)u(x,t)=\nu G (u-f),$$ with conditions $$\label{b} u_{t=0}=u_{t=b}, \,\, \int_X (u(x),1) \, dx =0$$ 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 $\Pi^2$.

• ### THEORY OF CATEGORIES

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.