What are the real-world implications of tangent vectors to a manifold?

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Discussion Overview

The discussion revolves around the real-world implications and interpretations of tangent vectors to a manifold, exploring their definitions and applications in various contexts such as general relativity, configuration spaces, and other mathematical frameworks. Participants examine the nature of curves on manifolds and the significance of tangent vectors in different physical and mathematical scenarios.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that tangent vectors can be understood as "velocity" vectors of curves on the manifold, prompting questions about the physical meaning of these curves.
  • Others argue that the interpretation of tangent vectors depends on the specific nature of the manifold, with examples including general relativity where vectors represent velocities of events.
  • A participant mentions that tangent vectors are derivatives and discusses their relationship to the tangent plane of a manifold.
  • Another participant introduces multiple definitions of tangent vectors, including those based on differential operators, differentiable curves, and coordinate specifications.
  • Some contributions highlight the importance of understanding the context of the manifold to make sense of tangent vectors, suggesting that different manifolds may yield different interpretations of curves and tangent vectors.
  • There is a discussion on the structure of vector bundles, emphasizing that each point of a manifold has an associated vector space, which leads to richer structures than simple vector spaces.
  • One participant contrasts the cylinder and the Möbius strip as examples of nonisomorphic vector bundles, illustrating the complexity of vector bundle structures.

Areas of Agreement / Disagreement

Participants express a variety of views on the nature and implications of tangent vectors, with no clear consensus reached. The discussion remains open-ended, with multiple competing interpretations and examples presented.

Contextual Notes

Some definitions and interpretations of tangent vectors depend on specific mathematical frameworks and assumptions, which may not be universally applicable across all contexts. The discussion also highlights the potential for different physical meanings associated with curves on various types of manifolds.

  • #31
Hi JD

One can look at your idea the other way around as well. Euclidean geometry is an abstraction of measurement on the Earth's surface, a curved manifold with irregular geometry, but which is approximately flat in small regions. From this point of view the curved manifold is the natural and concrete and the flat space is the axiomatically described abstraction.

The same is true of the Universe. It is a curved 4 dimensional surface that is approximately, in regions of small gravitational fields and velocities, a three dimensional Euclidean space that moves through absolute time.

regards

wofsy
 
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  • #32
Talking about the embedding of Whitney's theorem, it is written in Gallot, Lafontaine, Hullin, "But such an embedding is not canonical: the study of abstract manifolds cannot be reduced to the study of submanifolds of numerical space!"

What did they mean by that? What would it mean for two manifolds to be "naturally diffeomorphic" anyway?
 
  • #33
hi Quasar

The Whitney Embedding Theorem does not tell you how to construct an embedding of a manifold in Euclidean space but only that an embedding exists. The arguments involve approximation theorems and can not be directly applied to any specific example. In general there is no obvious way to embed an abstract manifold. each case must be taken separately.
That what it means to be non-canonical.
 

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