Re: Questions & Discussions on Vectors & relationships to di

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

Thanks David for your thoughtful reply. To continue the conversation:
> Regarding Joel's question:
>
> > Where does a position vector reside?
>
> First of all, it doesn't make much difference how you
> conceive of or draw
> the vectors when the manifold of interest is globally flat, since the
> manifold itself and its tangent space to it at each point are
> isomorphic.
> In such a case we can treat vectors in the usual way as
> having them reside
> in the space of interest (i.e. R^3).

This is what is tricky about e.g. R^3, its isomorphic to its tangent spaces;
but I'm not sure I want to say that it is the same as . . ., which is I
suppose why you said "make much difference" as opposed to no difference.

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."

for reasons that I won't go into, other than to say I'm trying to understand
affine transformations and spaces, I'm just now beginning to appreciate the
fact that the position is not a vector, its point in the manifold.  Years of
drawing arrows in 3d Euclidean space has led to my not appreciating this
fact.

> Regarding the specific question above, in the general curved
> space case,
> the position of a point in that curved manifold is *not* a
> vector at all.
> It is a point in the manifold.  The points in a manifold are not
> individually vectors.  Vectors live in vector spaces; curved manifolds
> are not vector spaces.  Since the position is not a vector, it doesn't
> reside as a vector anywhere.  It just resides as a point just where it
> is in the manifold.
>
> *But*, the *differential displacement* along a path between two
> infinitesimally nearby points in the manifold *is* a vector,
> and it lives
> in the tangent (vector) space of the point where that differential is
> taken.

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

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

Joel Rauber