Tangent Vectors (Part II)

The Tangent Bundle

In a previous post I discussed the notion of a tangent vector, the idea of an infinitesimal “push” at a point p \in M in a given differentiable manifold.  We measure or detect and, indeed, define, this push by testing how it acts on real-valued functions.  Using local coordinates (x_1, x_2, \dots, x_n) \in \mathbb{R}^n, these tangent vectors can be expressed as linear combinations of the operators

\displaystyle \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \dots, \frac{\partial}{\partial x_n}

all of which form a basis for the tangent space T_pM.  Because we often let the point p \in M be arbitrary — that is, we like to consider tangent vectors located at any tangent space T_pM on M — it is useful to gather or bundle all of these spaces together into one giant, unified set.  This new set is, perhaps not surprisingly, called the tangent bundle of the manifold M.

Def. (Tangent Bundle)  Let M be an n-dimensional real manifold.  The disjoint union of all the tangent spaces, T_pM, forms the tangent bundle.  It is notated as

\displaystyle TM = \amalg_{p \in M} \, T_pM

(The disjoint union symbol, \amalg, does, indeed, denote the same operation as the regular union symbol, \bigcup, except that it keeps the elements in each unioned set distinct from the elements in any of the other unioned sets.  If, for example, we drew M as a surface and visualized each T_pM as an actual plane in \mathbb{R}^3, then many of these tangent planes would overlap.  Forming a disjoint union allows us to keep track of overlapping points.  Another helpful comparison to consider is the standard union \mathbb{R} \bigcup \mathbb{R} vs. the disjoint union \mathbb{R} \amalg \mathbb{R}.  The first union results in the set \mathbb{R}, but the second one results in two distinct copies of the reals.)

The tangent bundle can be a rather tricky or overwhelming object to think about.  As a first step towards this end, its probably worth spelling out exactly what we mean when we say \omega \in TM:

\displaystyle \omega \in TM \iff \omega = (\vec{v}, p) \text{ where } \vec{v} \in T_pM.

Note, then, that there is a relationship between the vector part of \omega and the point part — the vector part, \vec{c}, is a tangent vector associated to the point, p.

Physcists have an excellent way of describing or thinking about this gigantic collection of spaces; they describe it as the set of all possible positions and velocities a particle traveling through the manifold M might have.

Interestingly enough, this tangent bundle can also be given the structure of a differentiable manifold.  Actually, my previous post should have pointed out that each individual tangent space, T_pM, can always be equipped with an n-dimensional differentiable structure, too.  Let’s first consider this and then discuss how we can extend these observations to TM.

A Differentiable Structure for Each T_pM

In order to think about each tangent space as a differentiable manifold, we need to supply it with coordinates.  Since every element \vec{v} \in T_pM can be expressed as

\displaystyle \vec{v} = a_1\frac{\partial}{\partial x_1} + a_2\frac{\partial}{\partial x_2} + \cdots + a_n\frac{\partial}{\partial x_n}

it follows that the n-tuple of reals, (a_1, a_2, \dots, a_n), completely encodes or captures \vec{v}.  Since \vec{v} \in T_pM is arbitrary, we see that an n-tuple of real numbers describes the entirety of T_pM.  That is, we arrive at a single (injective) function with a single (open) domain, U = \mathbb{R}^n, with which we can provide T_pM with a differentiable structure:

\displaystyle \vec{w}:\mathbb{R}^n\to T_pM \text{ is given by } \vec{w}(a_1, a_2, \dots, a_n) = \sum_{i=1}^n \! a_i \frac{\partial}{\partial x_i}.

There is only one transition function to check, T = \vec{w}^{-1}\circ\vec{w} = \text{id}, and so we are done.  And if you take a step back to contemplate this construction, you’ll probably realize that it was all very trivial.  At the end of the day, we’re really noting that any real vector space, V, can be provided with a differentiable structure so that V \cong \mathbb{R}^n.

  • Note: For reasons of notation, it can be a bit bizarre to contemplate tangent spaces of T_pM, but once this is sorted out, one realizes that they’re quite trivial, too.  Indeed, one discovers that T_{\vec{v}}\,T_pM \cong T_pM.
  • Note: In a few weeks we will have reason to think about each space, T_pM, as its own manifold.  For each p \in M, we will construct an important function called the exponential function \displaystyle \text{exp}_p : T_pM \to M, and we will argue that it is locally one-to-one, locally onto, and differentiable.  In order to make sense of this last adjective, we will need to equip the domain, T_pM, with a differentiable structure.

A Differentiable Structure for TM

The tangent bundle consists of pairs, (p, \vec{v}) where p \in M and \vec{v} \in T_pM.  Since we have n-dimensional local coordinates for p — the tuples (x_1, x_2, \dots, x_n) — and, as discussed above, we have n-dimensional local coordinates for \vec{v} — the coefficients (a_1, a_2, \dots, a_n) — it makes sense that the tangent bundle is a 2n-dimensional manifold.

To make this precise, given an arbitrary point (p, \vec{v}) \in TM, we need to specify a subset O containing (p, \vec{v}) and an injective function \vec{z}_{\alpha} : W_{\alpha} \to TM with \vec{z}_{\alpha}(W_{\alpha}) = O.  This turns out to be simpler than it may first appear.

We simply take W_{\alpha} = U_{\alpha} \times \mathbb{R}^n, where U_{\alpha} \subset \mathbb{R}^n is open and, thanks to the differentiable structure of M, comes equipped with an injective function \vec{x}_{\alpha} : U_{\alpha} \to M.  (Note that W_{\alpha} is an open set in \mathbb{R}^{2n} = \mathbb{R}^n \times \mathbb{R}^n.)  We then define \vec{z}_{\alpha} accordingly:

\displaystyle \vec{z}_{\alpha} ((x_1, \dots, x_n), (a_1, \dots, a_n)) = \Big{(} \vec{x}_{\alpha}(x_1, \dots, x_n), \, \sum_{i=1}^n a_i \frac{\partial}{\partial x_i} \Big{)}

Given two such functions, \vec{z}_{\alpha} and \vec{z}_{\beta}, we need to verify that the transition function T_{\alpha, \beta} = \vec{z}_{\beta}^{-1} \circ \vec{z}_{\alpha} is differentiable.  This, too, is surprisingly simple since we can first note that

\displaystyle T_{\alpha, \beta} = (\vec{x}_{\beta}^{-1} \circ \vec{x}_{\alpha}, d(\vec{x}_{\beta}^{-1}\circ\vec{x}_{\alpha}))

The first component is assumed to be differentiable since M was assumed to be a differentiable manifold, and the second component can be expressed as an n\times n matrix of differentiable functions.  As such, it is itself differentiable.  Hence, the entire transition function is differentiable.  (Read pages 15-16 for Do Carmo’s version of this.)

Here’s what is actually happening in plain language.  We are supposing that the point (p, \vec{v}) \in TM lives in two subsets, \vec{z}_{\alpha}(W_{\alpha}) and in \vec{z}_{\beta}(W_{\beta}).  We’re notating points from W_{\alpha} as pairs of n-tuples, $(x_1, \dots, x_n, a_1, \dots, a_n)$ with tangent vectors \vec{v} \in T_pM written using the a_i‘s as coefficients.  We need similar notation for points from W_{\beta}, and so we can agree to write them as $(y_1, \dots, y_n, b_1, \dots, b_n)$ where the b_i‘s tell us how to encode \vec{v} as an infinitesimal push in y_i coordinates.  That is

\displaystyle \vec{v} = a_1\frac{\partial}{\partial x_1} + \cdots + a_n\frac{\partial}{\partial x_n} = b_1\frac{\partial}{\partial y_1} + \cdots + b_n\frac{\partial}{\partial y_n}.

The a_i‘s are coefficients for \vec{v} expressed in the \partial/\partial x_i basis, while the b_i‘s are coefficients for \vec{v} expressed in the \partial/\partial y_i basis.  So how do we convert from one basis to another?  That’s right!  Via a change of basis matrix.  This change-of-basis matrix is precisely the differential of the mapping

\displaystyle \vec{x}^{-1}_{\beta} \circ \vec{x}_{\alpha} : U_{\alpha}

for which we have notation (see pages 9-10)

d\left(\vec{x}^{-1}_{\beta} \circ \vec{x}_{\alpha}\right): T_{q_{\alpha}} U_{\alpha} \to T_{q_{\beta}}

where q_{\alpha} and q_{\beta} are defined by \vec{x}_{\alpha}(q_{\alpha}) = p = \vec{x}_{\beta}(q_{\beta}).  In Vector Calculus we learn that given a differentiable function F: \mathbb{R}^n \to \mathbb{R}^m, its derivative matrix at q \in \mathbb{R}^n can be used to push vectors at q forward to vectors at F(q) — this is precisely what the differential of F does, too.  It uses this derivative matrix to push forward tangent vectors.

In short, then, the tangent bundle TM of a manifold M has a natural differentiable structure.  Points in this bundle consist of pairs (p, \vec{v}).  We have smooth transition functions that tell us how to express p in different coordinate patches, and, similarly, we have smooth transition functions that tell us how to express \vec{v} in different coordinate patches, too.

Examples

(1) Euclidean space.  If M = \mathbb{R}^n, then it necessarily follows that TM \cong \mathbb{R}^n\times\mathbb{R}^n \cong \mathbb{R}^{2n}.  Technically speaking, the tangent bundle is given by

T\mathbb{R}^n = \left\{ (p, \vec{v}) : p \in \mathbb{R}^n, \vec{v} \in T_p\mathbb{R}^n \right\}

but by making use of the one coordinate chart needed to define the standard differentiable structure on $\mathbb{R}^n$, this tangent bundle itself has a single coordinate chart, \vec{z} : \mathbb{R}^{2n} \to TM that serves as a diffeomorphism.

(2) The unit circle S^1 \subset \mathbb{R}^2.  In order to visualize TS^1, we place a copy of $latex \mat220px-Tangent_bundle.svghbb{R} \cong T_pS^1$ at every point p \in S^1, and we want to do so in a way so that distinct linear spaces T_{p_1}S^1 and T_{p_2}S^1 are disjoint.  This can be accomplished by drawing each linear subspace T_pS^1 as, say, a vertical line, yielding a cylinder.  In the image to the left, a failed attempt at doing this is shown at the top, with a successful attempt illustrated in the bottom.

(3) The unit sphere S^2 \subset \mathbb{R}^3.  The four-dimensional manifold TS^2 is, of course, impossible to visualize, but there is something else to note here. There is something about TS^2 that is fundamentally more than our first two examples.  For Eucldean spaces and the one-dimensional sphere we have TM \cong M \times \mathbb{R}^n; in other words, the tangent bundle was really nothing more than a product manifold (and this is the definition of a trivial bundle).  However, this does not hold for the unit sphere, that is TS^2 is not trivial.  A trivial tangent bundle means that at every point p \in Many tangent vector \vec{v} \in T_pM is allowed.  The Hairy Ball Theorem tells us that this is simply not the case for S^2.  To fully appreciate this we’ll need to think about vector fields, so stay tuned!

Oh, Why the Hell Not? Let’s Discuss General Bundles

Tangent bundles are but one example of a more general phenomenon.  Given a differentiable manifold M, it is possible to associate to every point p \in M other vector spaces, ones that “vary smoothly” from point to point, just like our tangent spaces T_pM do.  We use M‘s local coordinates to construct local coordinates for tangent vectors, too, and when we move through overlapping charts all of these coordinates change smoothly.

Similarly, we set up vector spaces and a notion of local coordinates so that something analogous to the tangent bundle situation happens; in fact, as in the case with our tangent bundle TM, we want the coordinates that parameterize “vector part” to not only transition smoothly but also to transition according to a linear function.  Wikipedia has a good description of this.

In Riemannian Geometry, we’ll need access to other vector bundles over M, too.  Probably the one we’ll (technically or secretly) use as much as the tangent bundle is the cotangent bundle.  Here’s a definition:

Definition (Cotangent Bundle).  Given a differentiable manifold M, the cotangent bundle is defined by

T^*M = \left\{ (p, \ell_p) : p \in M, \ell_p:T_pM \to \mathbb{R} \text{ is a linear function} \right\}

In other (simpler?) words, an element in T^*M consists of a point in M and a function that eats tangent vectors at said point.  (Hopefully this sounds a bit familiar, perhaps awfully familiar to differential forms.)  Just like the tangent bundle, T^*M is a 2n-dimensional manifold.

We will make use of other vector bundles over M, too, but we will be able to understand them without resorting to the general vector-bundle definition.  Indeed, we will understand them all locally and so many of the definitions will be hidden.

However, its worth setting up my next post on tangent vectors with the following observation: functions X: M \to TM (also called sections of TM) give us vector fields, while functions \omega : M \to T^*M (also called sections) give us differential 1-forms.

As we know from Spring’s Differential Geometry course, we will also want to define 2-forms, 3-forms and general n-forms.  What sort of vector space do we form at each p \in M to accomplish this?

 

 

 

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