# Riemannian Geometry

@article{HolopainenRiemannianG, title={Riemannian Geometry}, author={Ilkka Holopainen}, journal={Nature}, volume={119}, pages={117-117} }

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

#### 5,835 Citations

Riemannian problems with a fundamental differential system

- 2014

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 the… Expand

On the geometry of Riemannian manifolds with density

- Mathematics
- 2016

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 the… Expand

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

- Mathematics
- 2015

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 orientable… Expand

Ricci curvature and measures

- Mathematics
- 2009

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

- 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 relationship… Expand

Lipschitz algebras and derivations II: exterior differentiation

- Mathematics
- 1998

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

- 2018

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 homogenous… Expand

Applications of Affine and Weyl Geometry

- Mathematics, Computer Science
- Applications of Affine and Weyl Geometry
- 2013

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

- Mathematics, Physics
- 2015

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 defined… Expand

The moduli space of isometry classes of globally hyperbolic spacetimes

- Physics
- 2004

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 as… Expand

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