Riemannian functions and geometry

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Introduction

Einstein had to study Riemann math to be able to develop his EFE.

I decided to use this page to document all information I got about Riemann’s math.

I am using and working on the original Riemann document in German a document I imported into a Google doc.

index

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Riemann hypothesis

A Millenium prize is about this hypothesis.

 “the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function 

ζ(s) = 1 + 1/2s + 1/3s + 1/4s + …

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation

    ζ(s) = 0

lie on a certain vertical straight line.”

Read more in https://www.claymath.org/millennium-problems/riemann-hypothesis A good presentation is made by a “discovermats” Youtube channel mathematician:

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Riemann Zeta function

A nice presentation of Rieman hypothesis

With the help of wiki, I came to the original Riemann document in German. Riemann derivated the Zeta function from Euler’s product. I add information about Euler’s product in my equation page.

Here is  good presentation of this function.

( Source:  https://youtu.be/1jXNopn1Ivc )

( Source: https://www.youtube.com/watch?v=I3qSCWNXZKg )

Similar derivation made by Riemann from Euler’s product in the original Riemann document in German. :

( Image source: the original Riemann document. )

For p as all prime numbers and n as all whole numbers.

which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. ( Wiki )

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann‘s 1859 article “On the Number of Primes Less Than a Given Magnitude” extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.( Wiki )

The values of the Riemann zeta function at even positive integers were computed by Euler. ( Wiki )

The Riemann zeta function is connected to the Bernoulli numbers as in this image supplied by Mathologer:

(Source: Mathologer in www.youtube.com )

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Riemann 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 neighboring 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 neighborhoods 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 the 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 wiki page 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 the 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 open unit 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 develop this issue in a separate page at http://www.kinberg.net/wordpress/stellan/tensors/

 

 

 

A pluralist agnostic seeker

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