Yardsticks to Metric Tensor Fields

[fresh_42]

I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist, a classical geometer, an algebraist, a differential geometer, and a physicist “What is a metric?” then we get five different answers. I mean it is all about distances, isn’t it? “Yes” is still the answer and all do actually mean the same thing. It is their perspective that is different. This article is supposed to explain how.
The image shows medieval standards of comparison at a church in Regensburg, Germany, Schuh (shoe), Elle (ulna), and Klafter.

Hans Koberger – own work, CC BY-SA 4.0,

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[Ibix]

Reading through this slowly. Does a topologist really not care about the concept of angle? How do they define a "detour" or "in our direct way" without at least implicitly invoking the notion of an angle?

 

[fresh_42]

Reading through this slowly.

Thank you.

Does a topologist really not care about the concept of angle?

A metric in a metric space does not involve angles. E.g., the French railroad metric has no angles. At least I do not see any. Or the Manhattan ametric.

How do they define a "detour" or "in our direct way" without at least implicitly invoking the notion of an angle?

Per triangle inequality.

 

[Ibix]

A metric in a metric space does not involve angles. E.g., the French railroad metric has no angles. At least I do not see any. Or the Manhatten metric.
Per triangle inequality.

Right... I'm coming at this from a SR/GR perspective and may be equipping my spaces with more structures (and possibly more specific structures) than I should be. So if I understand you, a topologist does something like consider all possible routes from A to B and declare the shortest one(s) to be the direct route. Or the other way around - defines a procedure for describing the shortest route and derives a metric that respects that?

 

[fresh_42]

Right... I'm coming at this from a SR/GR perspective and may be equipping my spaces with more structures (and possibly more specific structures) than I should be. So if I understand you, a topologist does something like consider all possible routes from A to B and declare the shortest one(s) to be the direct route. Or the other way around - defines a procedure for describing the shortest route and derives a metric that respects that?

Yes, where "shortest" doesn't need to be Euclidean. You cannot shortcut routes in Manhattan through buildings, and you always have to run over Paris on French railroads.

 

[Ibix]

Thanks. And the classical geometer gets a notion of angle because he's slapped a distributive and associative inner product on top of the topologist's minimalism? He's also made a specific choice of metric (Euclidean), but given the inner product I think all that does is distinguish him from someone who studies more general Riemannian spaces.

 

[robphy]

To me (from the viewpoint of Cayley-Klein geometries),
an angle is a measure of separation between lines meeting at a point
as
a distance is a measure of separation between points joined by a line.
In a sense, they are "dual" to each other from this [projective and algebraic] viewpoint.
So, one may have to identify a notion of duality in more general metrics.

 

[WWGD]

Right... I'm coming at this from a SR/GR perspective and may be equipping my spaces with more structures (and possibly more specific structures) than I should be. So if I understand you, a topologist does something like consider all possible routes from A to B and declare the shortest one(s) to be the direct route. Or the other way around - defines a procedure for describing the shortest route and derives a metric that respects that?

Id say there are two main types of topology: Point set and Algebraic Point set eneralizes the properties of Euclidean Space to general Topological spaces. Algebraic Topology cosiders spaces, features up to continuous deformation, aka Homotopy, Isotopy (i.e., continuous deformations). But overall, I'd say, Topology is more about global properties : Compactness, Connectedness, etc , than local ones , such as angle, which are, I believe more geometric in nature. notice the 'metric' cognate in Geometry, derived from measurement. An angle may be defined for general matric spaces per Length Geometry / Metric Geometry. The book by BBI ( Burago, Burago Ivanov (https://www.amazon.com/dp/0821821296/?tag=pfamazon01-20) https://www.amazon.com/dp/0821821296/?tag=pfamazon01-20 ) describes such gehttps://www.amazon.com/Course-Metric-Geometry-Dmitri-Burago/dp/0821821296 ) Describes such generalizations to all metric spaces. Tere are too, so-called Length Spaces: Spaces containing allowable paths ( Such as the Manhattan Space, where you must use paths along a square grid), where you may generalize the concept of Geodesics, as the shortest ( per length function). There is a condition to determine such geodesics in Length Spaces. I can look it up if it interests you.

Another perspective: Ultimately, once you have an inner product <,> ( A map that assigns a Real number in [-1,1] ) to a pair of vectors in your space * ) in a vector space/ normed space, this also allows you to define an angle using:

$<a,b >=|| a|||| b||cos \theta$,
where ||.|| is a norm.

It's been a few years since I've read this book. Please let me know if I was unclear or skipped something*notice this is the full range of the values of cosine.

<a,b>