Spivak's notation re. tangent buldles (Diff. Geom. Vol. 1)

[Rasalhague]

Spivak: Diffrential Geometry, Vol. 1, p. 64:

To give this picture mathematical substance, we simply describe the 'arrow' from p to p+v by the pair (p,v). The set of all such pairs is just \mathbb{R}^n \times \mathbb{R}^n which we will also denote by T \mathbb{R}^n, the tangent space of \mathbb{R}^n; the elements of T \mathbb{R}^n are called 'tangent vectors' of \mathbb{R}^n. We will often denote (p,v) \in T \mathbb{R}^n by v_p ('the vector v at p'); in conformity with this notation, we will denote the set of all (p,v) for v \in \mathbb{R}^n by \mathbb{R}^n_{\enspace p}.

So a tangent vector is a pair (p,v); the tangent bundle T \mathbb{R}^n is the set of all such pairs, in this case, \mathbb{R}^{2n}.

But what is \mathbb{R}^n_{\enspace p}?

Is it

(1) A synonym for T \mathbb{R}^n, the tangent bundle;

(2) A subset of the tangent bundle: \pi^{-1}(\left \{ p \in \mathbb{R}^n: A(p)=1 \right \} \times \mathbb{R}^n), where A is some statement concerning elements of \mathbb{R}^n;

(3) The fibre over p: \left \{ p \right \} \times \mathbb{R}^n for some fixed p \in \mathbb{R}^n;

(4) A section with constant second component: \mathbb{R}^n \times \left \{ v \right \} for some fixed v \in \mathbb{R}^n?

On p. 66, he calls the expression on the left of c'(t)_{c(t)} \in \mathbb{R}^n_{\enspace c(t)} the tangent vector of c at t. From this, the right hand side could mean \pi^{-1}( c((-\varepsilon , \varepsilon)) \times \mathbb{R}^n), where (-e,e) is an open interval, and hence (2). But (3) seems reasonable also.

 

[klackity]

Spivak: Diffrential Geometry, Vol. 1, p. 64:



So a tangent vector is a pair (p,v); the tangent bundle T \mathbb{R}^n is the set of all such pairs, in this case, \mathbb{R}^{2n}.

But what is \mathbb{R}^n_{\enspace p}?

Is it

(1) A synonym for T \mathbb{R}^n, the tangent bundle;

(2) A subset of the tangent bundle: \pi^{-1}(\left \{ p \in \mathbb{R}^n: A(p)=1 \right \} \times \mathbb{R}^n), where A is some statement concerning elements of \mathbb{R}^n;

(3) The fibre over p: \left \{ p \right \} \times \mathbb{R}^n for some fixed p \in \mathbb{R}^n;

(4) A section with constant second component: \mathbb{R}^n \times \left \{ v \right \} for some fixed v \in \mathbb{R}^n?

On p. 66, he calls the expression on the left of c'(t)_{c(t)} \in \mathbb{R}^n_{\enspace c(t)} the tangent vector of c at t. From this, the right hand side could mean \pi^{-1}( c((-\varepsilon , \varepsilon)) \times \mathbb{R}^n), where (-e,e) is an open interval, and hence (2). But (3) seems reasonable also.

\mathbb{R}^n_{\enspace p} is the tangent space to \mathbb{R}^n at p. (The tangent space at a point p of any manifold M will always be isomorphic to \mathbb{R}^n, where n is the dimension of M at p. In this case, since M = \mathbb{R}^n, the tangent space at any point is actually isomorphic to M.)

The reason we need the tangent bundle in the first place is because initially, we don't have a way to relate \mathbb{R}^n_{\enspace p} and \mathbb{R}^n_{\enspace q} at two different point p and q. (Well, actually we do have a way to relate them in \mathbb{R}^n - namely translating the tangent vectors at different points to the origin. But we cannot do this for most manifolds, which is why we need the tangent bundle).

We'd like to be able to say things like, "This is a smooth vector field over \mathbb{R}^n". Since each of the vectors in a vector field over \mathbb{R}^n is attached to a different point, we need to know things like, "How close are two tangent vectors to each other that are attached to different points of the manifold \mathbb{R}^n"? The only way we can know this is if we put a topology on the space of ALL the possible tangent vectors attached to ALL the points of the manifold \mathbb{R}^n. This structure is what we call the tangent bundle.

When we construct the tangent bundle, we are actually using the spaces \mathbb{R}^n_{\enspace p} in the definition. We are joining, or bundling, all these tangent spaces together, and then putting a topology on it to make a new manifold. Although sometimes the tangent bundle ends up simply being diffeomorphic to M x \mathbb{R}^n, this is not always the case (which is why the definition of the tangent bundle is a bit more complicated than you might initially think it needs to be).

If you look at this wikipedia article: http://en.wikipedia.org/wiki/Tangent_bundle

T_xM refers to the same thing (the tangent space at a point x) as \mathbb{R}^n_{\enspace p} does in Spivak.

 

[Rasalhague]

Thanks, klackity. I've just been browsing that very page! I think I understand now. The tangent space at p, if I've got this right, is the preimage of the singleton {p} under the projection/submersion/fibration, T_pM = \pi^{-1}(\left \{ p \right \}), a.k.a. the fibre over p, so that, in this case, it's \left \{ p \right \} \times \mathbb{R}^n, my guess number (3).

 

[klackity]

Yes, (3) is a correct way to determine M_p given TM. But I think that's kind of putting the cart before the horse.

We know what M_p is before we even know what the tangent bundle is. M_p is defined to be the vector space of linear derivations at p (i.e., tangent space at p). So we can define M_p without any reference to TM at all.

However, we need to use M_p in the definition of TM.

I think I see now that T_pM is, by definition, the fiber over p. It is by virtue of the way TM was defined that T_pM \cong M_p.

So it is actually a special fact that M_p can be thought of as the fiber over p (i.e., T_pM), because M_p was not defined as such.

It is not a special fact that T_pM can be thought of as M_p (i.e., the tangent space at p), because TM was defined precisely in a way such that this fact would be true.

I hope you see the distinction I am making.

 

[Rasalhague]

I think so... It seems the distinction you're making is that the tangent bundle is defined in terms of tangent spaces, rather than the other way around. Is that it?

Incidentally, is "linear derivation" a tautology in the style of "linear vector space"? I thought linearity was part of the definition of a derivation (the other part being that it obeys the product/Leibniz rule).

 

[klackity]

I think so... It seems the distinction you're making is that the tangent bundle is defined in terms of tangent spaces, rather than the other way around. Is that it?

Incidentally, is "linear derivation" a tautology in the style of "linear vector space"? I thought linearity was part of the definition of a derivation (the other part being that it obeys the product/Leibniz rule).

Yeah, that's it. And also that T_pM and M_p are essentially the same object, but T_pM is defined as the fiber over p of TM, while M_p is defined as the vector space of derivations. They are the same thing, but defined differently, which has philosophical implications.

With its nice structure, the tangent bundle TM is now considered the natural object in which tangent vectors of M reside, instead of lone-wolf tangent spaces. So we like to use T_pM instead of M_p, because the elements of T_pM are by definition elements of TM. (The same is not true of elements of M_p -they are by definition derivations).And yes, I was taught the term "linear derivation", so I use that. But now that you mention it, the "linear" is not needed!