Riemannian Geometry

  title={Riemannian Geometry},
  author={Ilkka Holopainen},
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 theory of surfaces, and introduced certain fundamental ideas in this general theory. Bianchi, Beltrami, and others made substantial contributions to the subject, which was extended by Ricci with the use of tensor analysis and his absolute calculus. Recently there has been an extensive study and… Expand
Riemannian problems with a fundamental differential system
We introduce the reader to a fundamental exterior differential system of Riemannian geometry which arises naturally with every oriented Riemannian n+1manifold M. Such system is related to theExpand
On the geometry of Riemannian manifolds with density
We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as theExpand
On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume
We contribute to an original problem studied by Hamilton and others, in order to understand the behaviour of maximal solutions of the Ricci flow both in compact and non-compact complete orientableExpand
Ricci curvature and measures
Abstract.In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way,Expand
The fundamental group of non-negatively curved manifolds
  • D. Wraith
  • Mathematics
  • Irish Mathematical Society Bulletin
  • 1998
The aim of this article is to offer a brief survey of an interesting, yet accessible line of research in Differential Geometry. A fundamental problem of mathematics is to understand the relationshipExpand
Lipschitz algebras and derivations II: exterior differentiation
Abstract Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener space,Expand
Conformal Riemann Manifolds
As is well known F. Klein extracted the essence of the classical geometry by saying that the geometry is the study of properties invariant under the transformations of Lie groups on homogenousExpand
Applications of Affine and Weyl Geometry
This book associates an auxiliary pseudo-Riemannian structure of neutral signature to certain affine connections and uses this correspondence to study both geometries to be accessible to mathematicians who are not expert in the subject and to students with a basic grounding in differential geometry. Expand
Riemannian Geometry Based on the Takagi's Factorization of the Metric Tensor
The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well definedExpand
The moduli space of isometry classes of globally hyperbolic spacetimes
This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov–Hausdorff theory based on the use of the Lorentz distance asExpand


Partial Differential Equations
THE appearance of these volumes marks the happy conclusion of a work undertaken, as the author reminds us in his preface, twenty-one years ago. Doubtless it would have been finished earlier had itExpand
Sur les surfaces a courbure negative. (On surfaces of negative curvature )
  • C. R. Acad. Sci. Paris
  • 1926
Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. (German). (The asymptotic distribution law of the eigenvalues of linear partial differential equations)
  • Math. Ann
  • 1912
2 Induced metric on surfaces
  • 2 Induced metric on surfaces
A proof of the well conjecture using Ricci flow
  • A proof of the well conjecture using Ricci flow
A semi-discrete, linear curve shortening flow
  • American Math. Monthly
Aleksandrov spaces with curvatures bounded from below
  • Aleksandrov spaces with curvatures bounded from below
Asymptotic stability of Ricci pseudosolitons
  • Asymptotic stability of Ricci pseudosolitons
Classes of domains and imbedding theorems for function spaces
  • Classes of domains and imbedding theorems for function spaces
Compact Ricci solitons
  • Compact Ricci solitons