Yardsticks to Metric Tensor Fields

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fresh_42
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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.
360px-Regensburg_-_Altes_Rathaus_-_Masse_-_2016.jpg
Hans Koberger – own work, CC BY-SA 4.0,

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Ibix
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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?
 
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fresh_42
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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.
 
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Ibix
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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?
 
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fresh_42
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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.
 
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Ibix
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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.
 
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robphy
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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.
 

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