Manifold #9. The Tangent Space - Definition매니폴드 #9. 접공간의 정의

These are my notes on XylyXylyX’s What is a Manifold? lectures. This post covers lesson 9 of the playlist, The Tangent Space - Definition. The earlier material is in the previous post.

Why build the tangent space again?

In the Tensor series we declared that we put coordinates on spacetime and, at each point, erect a vector space, its dual space, and the tensor product spaces. The basis of that vector space was the differential operators of the coordinate system. That declaration was enough for learning tensors, but some parts were stated without a foundation. Now that we know manifolds, we can build that vector space directly from the structure of a manifold. This lecture is among the hardest in the series, so it matters not to lose the order in which the definitions are built.

We start from the manifold so far. We do not say spacetime or four-dimensional, and we simply call it $X$, with a topology $\mathcal{T}_X$ and an atlas.

The collection of smooth functions

Take a function $f : X \to \R$ from the manifold $X$ to the real line $\R$. The target $\R$ is already a Euclidean space, so we need no extra chart. It is as if we put on $\R$ an atlas with a single chart, the identity. So the smoothness of $f$ is decided by one chart $\gamma$ of $X$.

If this function is infinitely differentiable, we call $f$ a smooth function, and write the set of all such functions as $C^\infty(X)$.

$C^\infty(X)$ is a vector space. Define the sum of two functions pointwise and the real multiple pointwise, and sums and real multiples of smooth functions are again smooth, so the set is closed under both operations. This fact will connect later with the point that a velocity is a map that eats functions.

From curves and functions to velocity

Let us reuse the curve $\psi : \R \to X$ from the previous post. We write the parameter as $t$, and it is handy to think of it as time. $\psi(t_0)$ is a point of $X$. Given a curve $\psi$ and a function $f$, composing them gives $f \circ \psi : \R \to \R$, an ordinary function from one real number to one real number, so we can differentiate it. It measures how fast the value of $f$ changes as we move along the curve.

In a picture it goes like this. The points where $f$ takes the same value form a contour of $f$, and drawing these contours on the manifold gives the manifold a contour pattern. How fast the curve crosses these contours as we follow it is $(f \circ \psi)'(t_0)$. This value depends strongly on the curve. Even through the same point, a different curve crosses the contours differently.

A word on notation. Many books write this derivative as $df/dt$, but the domain of $f$ is $X$, not the line of $t$. Here $d$ has nothing to do with the differential forms of earlier lectures, and what is really differentiated is $f \circ \psi$. If we give it a new name such as $\tilde{f}(t) = f(\psi(t))$, everything fits. Many books use this shorthand without explanation, so when reading it is good to check always which composition $f$ stands for.

Q. Why can we not simply write $df/dt$?
A. $f$ is a function on $X$, so it does not take $t$ as input. The function that takes $t$ is $f \circ \psi$, which first passes through the curve, and $df/dt$ is only a shorthand for it.

Definition of velocity

Now we build a new object. Fix a curve $\psi$ and a parameter value $t_0$, and define a map $v_{\psi, t_0} : C^\infty(X) \to \R$, which takes a function and gives a real number, as follows.

Compose the curve with the function $f$, differentiate in $t$, and read the value at $t_0$. We call this map the velocity of the curve $\psi$ at $t_0$. It is better understood as the velocity the curve has relative to the contours of $f$ than as a velocity through spacetime. The definition needs both the curve and a point on it, so it carries two subscripts.

Once we fix the point $P$ with $\psi(t_0) = P$, countless curves pass through $P$. A curve $\phi$ may pass through $P$ at $t_1$ and a curve $\sigma$ at $t_2$, and their velocities $v_{\psi, t_0}$, $v_{\phi, t_1}$, $v_{\sigma, t_2}$ give different real numbers for the same $f$. This is because the curves explore $f$ in different ways. A few curves may happen to give the same value. No curve is special, so there is no reason to pick just one and define something like a tangent line from it.

Gather all the velocities

When no curve is special, the remedy is to collect all of them. When we built the dual space there was likewise no reason to pick one map, so we collected all possible maps. In the same way, we write the set holding the velocities of all curves through the point $P$ as $T_P X$ and call it the tangent space at $P$. The elements of this set are maps from $C^\infty(X)$ to $\R$.

To show that $T_P X$ is a vector space, we must check several things. In this post we prove only the trickiest, closure under addition. When we pick two velocities $v_{\psi, t_0}$ and $v_{\phi, t_1}$, their sum must again be the velocity of some curve through $P$. The sum is defined like this. Feed in a function $f$ and the two velocities give two real numbers, and we take their sum. So we must find a curve $\Sigma$ for which $v_{\Sigma, 0}(f)$ equals this sum for every $f$. What remains after erasing $f$ is the addition of the tangent space.

Besides addition, scalar multiples must also be closed. Stretching the parameter of a curve by a factor $a$ to make $\psi(t_0 + a t)$ multiplies the velocity by $a$, so a velocity multiplied by a real number $a$ is again an element of the tangent space. A constant curve gives the zero vector, and $a = -1$ gives the opposite velocity. (This explanation is added to aid understanding and was not covered in the lecture.)

The sum curve Σ

We start in one chart $(U, \gamma)$. The proof does not depend on the choice of chart, but we need one chart to make our way through. The curves $\psi$ and $\phi$ pass through $P$, so $\psi(t_0) = \phi(t_1) = P$. We define the sum curve $\Sigma$ by adding and subtracting in chart coordinates.

Everything inside the parentheses is an element of $\R^d$, so it can be added and subtracted. The outer $\gamma^{-1}$ lifts the result back to the manifold. Points cannot be added on the manifold, so we must go through chart coordinates. This formula makes sense when $t$ is small enough near 0 that the coordinates inside the parentheses stay in $\gamma(U)$. (This explanation is added to aid understanding and was not covered in the lecture.)

First let us see whether $\Sigma$ passes through $P$. Putting in $t = 0$, we have $\gamma(\psi(t_0)) = \gamma(\phi(t_1)) = \gamma(P)$, and the inside is $\gamma(P) + \gamma(P) - \gamma(P) = \gamma(P)$. So $\Sigma(0) = \gamma^{-1}(\gamma(P)) = P$. This curve passes through $P$ at $t = 0$.

The formula seems to treat $\psi$ specially, but we could use $\phi$ in the last term. That term’s job is to subtract the coordinates $\gamma(P)$ of the point $P$ once, so it gives the same value whichever curve we use.

In the toy below, drag the velocity arrows of the two curves. You can check with numbers that the velocity of the orange sum curve $\Sigma$ is the sum of the two velocities. This is the simplest case, with the chart left as the plane coordinates, and the parameters of the curves are set so that each passes through $P$ at $t = 0$. (This toy is added to aid understanding and was not covered in the lecture.)

Q. Why do we not add the curves directly on the manifold, and go through chart coordinates?
A. A manifold is a set of points, so addition of points is not defined on it. Addition exists in $\R^d$, so we bring the coordinates down with the chart, add, and lift back up with $\gamma^{-1}$.

The velocity of the sum curve

Now we compute $v_{\Sigma, 0}(f) = (f \circ \Sigma)'(0)$. First insert $\gamma^{-1} \circ \gamma$ and write $f \circ \Sigma = (f \circ \gamma^{-1}) \circ (\gamma \circ \Sigma)$. We inserted the identity map, so the value is unchanged, but this lets us see the domain and codomain of each piece.

• $g = f \circ \gamma^{-1}$ is a function from $\R^d$ to $\R$.
• $\alpha = \gamma \circ \Sigma$ is a curve from $\R$ to $\R^d$, and by the definition the $\gamma^{-1}$ disappears, giving $\alpha(t) = \gamma(\psi(t_0 + t)) + \gamma(\phi(t_1 + t)) - \gamma(P)$.

So what we want is the derivative of the composition $g \circ \alpha$, which goes from $\R$ through $\R^d$ to $\R$. We use the multivariable chain rule. The partial derivatives of the outer function must be measured at the point the inner function delivers, which here is $\alpha(0) = \gamma(P)$.

The $i$-th component of $\alpha$ is the sum of the $i$-th coordinates of the two curves minus the constant $\gamma^i(P)$. A constant disappears under differentiation, and the derivative of a sum is the sum of derivatives, so we have this.

Put this into the formula above and split into two sums.

The first sum has the shape of the right side of the chain rule read backwards. Since $g \circ \gamma \circ \psi = f \circ \psi$, the first sum is $(f \circ \psi)'(t_0)$, and by the same reason the second sum is $(f \circ \phi)'(t_1)$. The conclusion is this.

Since $f$ was an arbitrary element of $C^\infty(X)$, we can erase $f$ and write $v_\Sigma = v_\psi + v_\phi$. We have shown that the sum of two velocities lies in the tangent space again, as the velocity of the curve $\Sigma$ through $P$, and this is the addition of the tangent space. The chart $\gamma$ used during the proof does not remain in the conclusion. Curves of different shapes can have the same velocity, so whichever chart we use, the result is the same.

Q. We used a chart in the proof, so does the result not depend on the chart?
A. It does not. The chart is a tool inserted to use the chain rule, and only curves and functions remain in the last formula. The coordinate expressions of the curve look different in different charts but give the same velocity.

Summary

The velocities of all curves through a point $P$, that is, the collection of maps $f \mapsto (f \circ \psi)'(t_0)$ from $C^\infty(X)$ to $\R$, form a vector space. This is the tangent space $T_P X$ at $P$. Today we proved only closure under addition, and scalar multiples, the zero vector, and opposite vectors can be checked in a similar way. In the next lecture we look further at what the tangent space is used for.