Riemannian geometry

“Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds”(Source: Wiki )

Manifolds

“The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space.” So let us look closer at the definition oand examples of manifolds. I will borrow texts from Wiki that I find are clear enough.

(In Swedish Mångfald, in Italian: Varietà) is a topological space that locally resembles Euclidean space (can be located by three coordinates, x, y, z)  near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.” (Source: Wiki )

One-dimensional manifolds
include lines and circles, but not figure eights 

Two-dimensional manifolds
are also called surfaces. Examples include the plane, the sphere, and the torus (like a donut) and the Klein bottle, which can all be embedded (formed without self-intersections) in three dimensional real space

The surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

Topology

In mathematicstopology (from the Greek τόπος, ‘place’, and λόγος, ‘study’) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. (Source: Wiki )

topological space may be defined as a set of points, along with a set of neighbourhoods for each point, (Source: Wiki )

Riemannian geometry

Back to the definition I started this page with: 

“Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds” (Source: Wiki )

“Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.”

(Source: Wiki )

The English wiki doesn’t have always the easiest explanation so I rely on the Swedish 

I started with https://sv.wikipedia.org/wiki/Differentialgeometri 

With this Swedish wikipage I found the site http://mathworld.wolfram.com/

that has several good explanations.

“A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x,y) is defined as the length of the shortest curve (geodesic) between x and y.”

I like the definition of manifold in worlfram.com

“A manifold is a topological space that is locallyEuclidean (i.e., around every point, there is a neighborhood that is topologically the same as the openunit ball in R^n). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly “flat” on small scales is a manifold” (Source: mathworld.wolfram.com/Manifold.html )

Reading more about differential geometry I soon need to understand tensors. I develope this issue in a separate page at http://www.kinberg.net/wordpress/stellan/tensors/

 

 

 

A agnostic pluralist seeker

Insert math as
Block
Inline
Additional settings
Formula color
Text color
#333333
Type math using LaTeX
Preview
\({}\)
Nothing to preview
Insert