• Corpus ID: 1224658


  author={Paul Douglas Lockhart},
This paper is an introduction to Riemannian and semi-Riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames, curvature, geodesics and homogeneity on six model spaces: Rn, Sn, and Hn in Riemannian geometry, and Rn,1, dSn and AdSn in semiRiemannian geometry. We then claim that up to isometries, these are the only examples in both settings, and state a classification theorem to support that. Some terminology from general relativity will be introduced… 
An Intrinsic Formula for the Cross Ratio in Spherical and Hyperbolic Geometries
Summary The cross ratio is a well known and useful invariant in Euclidean and projective geometries. Since it is invariant under Möbius transformations, it extends to spherical and hyperbolic
On the role of non-local Menger curvature in image processing
The Menger curvature is presented, which goes beyond classical curves and Riemannian manifolds to general metric spaces and is rigorously defined on a variety of discrete settings and allowed to be computed in a robust manner.
Singular vertices of nonnegatively curved integral polyhedral 3-manifolds
In this paper, we study polyhedral 3-manifolds with nonnegative curvature and integral monodromy, two conditions motivated by Thurston’s work in [Thu98]. We classify the 32 isometry types of
Assuming knowledge of Euclidean geometry, metric spaces, and simple analysis, I introduce some tools from differential geometry in the world of two-dimensional Euclidean space. I then apply these
ABsTRAcr. The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real
Fundamental Domains in Lorentzian Geometry
We consider discrete subgroups Γ of the simply connected Lie group S̃U(1, 1) of finite level, i.e. the subgroup intersects the centre of S̃U(1, 1) in a subgroup of finite index, this index is called
Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds
The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in C , and has a natural coordinate system (introduced by Linda Keen and Caroline
Mutually Tangent Spheres in n-Space
Summary In this note we prove that the points of tangency of n + 1 mutually tangent spheres in n-dimensional space lie on a generalized sphere. Coxeter's observation that for each of five mutually


Riemannian Manifolds: An Introduction to Curvature
What Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.-
Semi-Riemannian Geometry With Applications to Relativity
Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries.
Elementary Differential Geometry
Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of
Differential forms and applications
1. Differential Forms in Rn.- 2. Line Integrals.- 3. Differentiable Manifolds.- 4. Integration on Manifolds Stokes Theorem and Poincare's Lemma.- 1. Integration of Differential Forms.- 2. Stokes
Algorithms in Combinatorial Geometry
  • H. Edelsbrunner
  • Mathematics
    EATCS Monographs in Theoretical Computer Science
  • 1987
This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Computational Geometry
  • F. F. Yao
  • Mathematics
    Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity
  • 1990
Differential Geometry, Publish or Perish
  • Differential Geometry, Publish or Perish
  • 1979
and M
  • I. Shamos, Computational Geometry, Springer-Verlag, New York,
  • 1985