Re: Questions & Discussions on Vectors & relationships to di

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Joel's comments & questions:
> ...
> In reference to my question about "position vectors" and your response
> below.  Which is what I guessed the answer would be.  Would you say that
> this one difference, that even in flat space the position "is *not* a vector
> at all."

It is true that, technically speaking, the position is not a vector in an
ordinary flat space any more than it is in a curved one, but in a flat
space we can easily put the positions into a 1-1 correspondence with each
of the vectors of the tangent space that is tangent to the original
space's origin when it is described in Cartesian coordinates.

Consider a flat space whose points are labeled by Cartesian coordinates,
and consider the tangent space tangent to it at its origin.  Because the
space is flat the geodesic displacement from the origin to each of the
individual points in the space is in a 1-1 correspondence with each of
the points themselves.  Also, each of the vectors of the tangent space
can be thought of as a superposistion of the coordinate basis vectors
whose components are the actual finite physical displacements along each
of the coordinate directions necessary to arrive at each point of the
space when starting from the origin. Such finite displacements can be
put into a 1-1 correspondence with the vectors of the tangent space
because the flat space itself remains coincident with the tangent space
everywhere rather than just in an infinitesimal neighborhood of the point
of tangency (i.e. the origin in this case).

> ...
>
> Am I correct in saying that tangent spaces are necessarily Flat?

Yes.  The tangent spaces are ordinary Cartesian vector spaces.

> But are they necessarily Euclidean?  (I'm guessing "no")?

Certainly we can endow the tangent spaces with an indefinite pseudo-
Euclidean form for the metric interval (such as the Minkowski metric
interval of SR) that it can inherit from the original manifold.  But that
is the only kind on non-Euclideanness that can be allowed in the tangent
spaces.  Since the tangent spaces are ordinary real-valued vector spaces,
their vectors are ordinary real Cartesian-type superpositions of their
basis vectors. (Geometrically, the tangent spaces are just R^n.)

David Bowman
David_Bowman@georgetowncollege.edu