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

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Tangent vectors to a manifold can be understood as "velocity" vectors of curves on that manifold, with their interpretation varying based on the nature of the manifold's points. For instance, in general relativity, tangent vectors represent velocities of spacetime events, while in Euclidean space, they correspond to particle trajectories. The discussion highlights various definitions of tangent vectors, including those based on differential operators, curves, and coordinate systems. Additionally, the concept of tangent bundles is explored, emphasizing that each point on a manifold has an associated vector space, leading to a richer structure than simple vector spaces. Understanding tangent vectors requires considering the physical meaning of curves on the manifold and how they relate to the manifold's specific context.
  • #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 arguements 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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