# 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…

## 7,098 Citations

### Riemannian problems with a fundamental differential system

- Mathematics
- 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…

### 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…

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

- Mathematics
- 2015

### The Geometry of the Frame Bundle over Spacetime

- Mathematics
- 2000

One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The…

### IX.—On the Line-Geometry of the Riemann Tensor

- MathematicsProceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences
- 1944

In this paper the curvature tensor Rijkl in a Riemannian Vn is used to define a quadratic complex of lines in an (n – I)-dimensional projective space Sn–1. Work in this direction has been done for a…

### The fundamental group of non-negatively curved manifolds

- MathematicsIrish 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…

### 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,…

### Applications of Affine and Weyl Geometry

- MathematicsApplications 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.

### The moduli space of isometry classes of globally hyperbolic spacetimes

- Mathematics
- 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…

### Some topics in differential geometry of normed spaces

- MathematicsAdvances in Geometry
- 2020

Abstract For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in…

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