Friedmann cosmology and almost isotropy

  title={Friedmann cosmology and almost isotropy},
  author={Christina Sormani},
  journal={Geometric \& Functional Analysis GAFA},
  • C. Sormani
  • Published 19 February 2003
  • Mathematics
  • Geometric & Functional Analysis GAFA
AbstractIn the Friedmann model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur’s theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally isotropic in a sense which allows both weak gravitational lensing in all directions and strong gravitational lensing in localized angular regions at most points. We then… 
Se p 20 07 Conjugate Points in Length Spaces
In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and
Conjugate points in length spaces
Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space
The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this
Scalar Curvature and Intrinsic Flat Convergence
Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov
Stability of graphical tori with almost nonnegative scalar curvature
By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori
How Riemannian Manifolds Converge: A Survey
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We
How Riemannian Manifolds Converge
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We
In 1991, Gromov introduced the Gromov-Hausdorff distance between compact Riemannian manifolds. Applying the Bishop-Gromov Volume Comparison Theorem, he proved that sequences of Riemannian manifolds,
Intrinsic Flat Arzela-Ascoli Theorems
One of the most powerful theorems in metric geometry is the Arzela-Ascoli Theorem which provides a continuous limit for sequences of equicontinuous functions between two compact spaces. This theorem
Mathematical Aspects of General Relativity
Mathematical general relativity, the subject of this workshop, is a remarkable confluence of different areas of mathematics. Einstein's equation, the focus of mathematical relativity, is one of the


Stability problems in a theorem of F. Schur
Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of
On the proof of the positive mass conjecture in general relativity
LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's
Hausdorff convergence and universal covers
We prove that if Y is the Gromov-Hausdorff limit of a sequence of compact manifolds, M n i , with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then Y has a
Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics
The primary purpose of all physical theory is rooted in reality, and most relativists pretend to be physicists. We may often be members of departments of mathematics and our work oriented towards the
Ricci curvature and volume convergence
The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the
Spheres with locally pinched metrics
The purpose of this paper is the construction of metrics on spheres whose curvatures are locally almost constant but which have large variation globally. This construction also applies to spherical
Black holes and the Penrose inequality in general relativity
In a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least
Gravitational Curvature – An Introduction to Einstein's Theory
Theodore Frankel 1979 San Francisco: W H Freeman xviii + 172 pp price £5.20 (paperback) This book, although subtitled 'An introduction to Einstein's theory', is, as the author admits, not suitable
Principles of Physical Cosmology
This book is the essential introduction to this critical area of modern physics, written by a leading pioneer who has shaped the course of the field for decades. The book provides an authoritative