Tangent Bundle: Why It's Important

[observer1]

Good Morning All:

I am now understanding a bit -- just a bit: still struggling - about the tangent bundle.

But I have no idea WHY this is important.

As I understand, at every point on a manifold (or, more appropriately: at the coordinates placed on a manifold by a mapping), we study the union of the POINT and the TANGENT SPACE for all points. I sort of get that.

(I do not know why we make a thing of it, though.)

But then they go into this idea of TWO sub-manifolds: 1) fiber and 2) base.

And then things get hazy: it looks like the base is the space of all tangent vectors that interesect with tangent vectors from other points. And the fiber is the non-intersecting components of the tangent vectors.

I think. Now I am not so much asking for clarification on the tangent bundle (although I would not mind any clarifications).

*** I am asking: WHY ARE WE DOING THIS? *** What is to be gained? I SORT of think it might have to do with position vectors and momentum vectors (in physics, at least).

But NO book I see is even taking the time to say WHY we are splitting this up... if we are... and WHY we need to do it for all the points in a union (does that justify the "CONNECTION" in the covariant derivaitve?)

Please try not to be too mathematical if you are so kind as to respond. I am flinging words around here and not entirely so sure I know what I am talking about.

 

[RockyMarciano]

It's not that you "need to" use it, it simply gives you more versatility. I believe the whole idea(and I'm referring here to the more general idea of fiber bundle of which the tangent bundle is just an example) is about generalizing the topological product, by making it a trivial case of the fiber bundle. As all generalizations this allows more flexibility to model different things by introducing more possible distinctions. I don't know in which context you are studying it, I could see how in a purely abstract mathematical setting it could be hard to find it purpose if you aren't abstract minded, certainly in physics it has been fruitful and the two principal theories, GR and QFT benefit from the bundle view. It also offers an interesting take on classical mechanics.

 

[strangerep]

As I understand, at every point on a manifold (or, more appropriately: at the coordinates placed on a manifold by a mapping), we study the union of the POINT and the TANGENT SPACE for all points.

For later use, let's call the base manifold $\mathbb M$ and assume we have the usual notion of coordinates $x^\lambda$ on (at least an open subset of) $\mathbb M$. In physics we are typically interested in curves on $\mathbb M$. Generically, let's denote the coordinates of an arbitrary curve by $\gamma^\mu(s)$, where $s$ is an arbitrary (affine) parameter. For each point $x$ through which $\gamma(s)$ passes, its tangent vector at $x$ may be defined as $$u^\mu(x) ~:=~ \left. \frac{d\gamma^\mu(s)}{ds} \right|_x ~~.$$The set of all possible tangent vectors at $x \in {\mathbb M}$ is known as the tangent space at $x$, denoted $T_x{\mathbb M}$. The tangent bundle of ${\mathbb M}$ is defined as the union $$T{\mathbb M} ~:=~ \bigcup_{x \in {\mathbb M}} T_x{\mathbb M} ~~.$$

(I do not know why we make a thing of it, though.)

The point of this is just to establish a mathematical framework in which we can talk about points on a base manifold, and all the possible curves through any point on the manifold. We can talk about the base points $x$, together with possible directions at $x$.

*** I am asking: WHY ARE WE DOING THIS? *** What is to be gained? I SORT of think it might have to do with position vectors and momentum vectors (in physics, at least).

Yes, that's one type of usage: symplectic phase space in classical mechanics. But it's more widely useful than that, because it encapsulates an important distinction between certain physical properties of a system, known (in mathematics) as a vertical-horizontal decomposition: features describable at each point of ${\mathbb M}$ separately are called "vertical", whereas features that require a neighborhood in ${\mathbb M}$ for their description are called "horizontal". In other words, vertical features pertain to each tangent space. When combined with horizontal features, we recover the complete set of features on the tangent bundle.

Examples of a vertical features are spin orientation, or velocity direction, whereas a horizontal feature would be something like relative spatio-temporal displacement between 2 events.

But NO book I see is even taking the time to say WHY we are splitting this up... if we are... and WHY we need to do it for all the points in a union (does that justify the "CONNECTION" in the covariant derivative?)

To be physically useful, we want the vertical-horizontal decomposition to be independent of coordinates on the tangent space. Under a change of coordinates, vertical features must stay vertical and horizontal features must stay horizontal. This leads to an abstract notion called an Ehresmann connection, which specifies how vertical/horizontal are to be delineated. The more familiar "affine connection" in Riemannian geometry is a special simplified case.

To explain much further, I'd need to introduce a lot more math. But I'm not sure of your background.