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

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SUMMARY

Tangent vectors to a manifold are fundamentally defined as derivatives and can be interpreted as velocity vectors of curves on the manifold. For instance, in the context of general relativity, tangent vectors represent velocities of spacetime events. Spivak outlines multiple definitions of tangent vectors, including those based on differential operators, differentiable curves, and coordinate specifications. The discussion emphasizes the importance of understanding the nature of the manifold's points to grasp the physical meaning of curves and their associated tangent vectors.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly tangent bundles.
  • Familiarity with Spivak's definitions of tangent vectors and bundles.
  • Knowledge of differentiable manifolds and their properties.
  • Basic grasp of vector calculus and its applications in physics.
NEXT STEPS
  • Study the concept of tangent bundles in differential geometry.
  • Explore Spivak's definitions of tangent vectors in detail.
  • Learn about the physical interpretations of curves on manifolds in various contexts.
  • Investigate the differences between isomorphic and non-isomorphic vector bundles.
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Mathematicians, physicists, and students of differential geometry seeking to deepen their understanding of tangent vectors and their applications in various fields, including general relativity and theoretical physics.

  • #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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