Negative curvature in low dimensions

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Discussion Overview

The discussion revolves around the concept of negative curvature in low-dimensional spaces, specifically focusing on the nature of curves and their properties. Participants explore the implications of curvature in mathematical contexts, with references to specific types of curves such as hyperbolas and parabolas.

Discussion Character

  • Exploratory, Debate/contested, Conceptual clarification

Main Points Raised

  • One participant suggests that a hyperbola may represent negative curvature but expresses confusion about how this curvature manifests, comparing it to two parabolas.
  • Another participant asserts that curves can indeed have negative curvature, indicating a general acceptance of this concept.
  • A separate contribution discusses the relationship between positive and negative curvature, suggesting that if a function has positive curvature, its negative can be derived by inverting the function.
  • One participant expresses a desire for clarity and understanding, questioning the need for a complete case and emphasizing their interest in the broader context of the universe rather than seeking definitive answers.
  • There is a mention of potential misunderstanding or miscommunication regarding the expectations of the discussion, highlighting the challenges of conveying complex ideas in a forum setting.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the nature of negative curvature, with some expressing confusion and others affirming its existence. The discussion remains unresolved with competing viewpoints on how to interpret and understand negative curvature in the context presented.

Contextual Notes

There are indications of missing assumptions regarding the definitions of curvature and the specific types of curves being discussed. The conversation reflects varying levels of understanding and communication styles among participants.

Paige_Turner
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TL;DR
Can a plane curve have negative curvature?
It's probably the hyperbola, but I don't see how it's curvature is negative. It looks like 2 parabolas.
thanx, paige turner
 
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Hello and :welcome: !

You may be able to see what "it" looks like, but we are mere humans and not telepathic.
Pleas present a complete case.

And if all you want is an answer: yes, curve can have a negative curvature.

##\ ##
 
If f(x) has positive curvature, then -f(x) has negative curvature. It is possible for a curve to curve downward
 
BvU said:
Hello and :welcome: !

> You may be able to see what "it" looks like, but we are mere humans and not telepathic.
Well, I'm autistic--not human at all. Ask my friends. The subject line indicates that "it" is a plane curve with negative curvature.

> Please present a complete case.

A case for what? I don't have an agenda.I don't have the answer. I just want to understand WITW is going on around me, and what's "around me" is the universe.

> if all you want is an answer

Um... what else could I want? I asked a question.

I just got here and I'm already in trouble with the mods. If past is indeed prologue, I will be banned and not have the slightest idea why--even when I'm polite and only talk about hypergeometry.
 
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