Riemannian connections

In course

Description

This lecture covers the concept of Riemannian connections on manifolds, defining connections as maps that relate tangent vectors on a manifold. It explores the properties and operations of these connections, emphasizing their smoothness and compatibility with the metric. The lecture also delves into the Leibniz rule and the symmetric nature of connections. Additionally, it discusses the unique Riemannian connection, known as the Levi-Civita connection, which is both symmetric and metric-compatible. The presentation concludes by highlighting the existence of infinitely many connections on a manifold and the significance of these connections in Riemannian geometry.

Official source

https://mediaspace.epfl.ch/media/0_zwtlunjt

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Related concepts (64)

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

Definition

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Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.

Lexical definition

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