Tangent Space

Now we define tangent vectors on a manifold and the space they generate, the tangent space.

Motivation

In [Lee], a fairly lengthy portion of the beginning of Chapter 3 is devoted to justifying the definitions we will make; let us first briefly review this.

The simplest example of a manifold is \(\mathbb{R}^m\), and a slightly more complicated example is a surface contained in \(\mathbb{R}^m\). A tangent vector at a point \(p\) on this surface literally means vectors tangent to the surface at \(p\). If this surface is a two-dimensional surface in \(\mathbb{R}^3\), then collecting these vectors yields exactly the tangent plane at \(p\). However, generalizing this to a definition of tangent vectors on a manifold is not straightforward. In this definition, the existence of an ambient space \(\mathbb{R}^m\) containing the surface is essential, because we defined a manifold in a purely intrinsic way.

Instead, in the situation above we can observe that directional derivatives arise whenever tangent vectors are given. That is, given any tangent vector \(v\), this vector well-defines the directional derivative in the \(v\)-direction

\[\lim_{t\rightarrow 0}\frac{f(p+tv)-f(p)}{t}\]

for each function \(f\) defined in a neighborhood of \(p\). Our idea is to define this operator, the directional derivative, as the tangent vector.

Sheaf of Differentiable Functions

Let \(C^\infty(U)\) be the collection of \(C^\infty\) functions defined on an open set \(U\), and whenever \(V\subseteq U\) define the map \(\rho_{UV}:C^\infty(U)\rightarrow C^\infty(V)\) by

\[\rho_{UV}:f\mapsto f\vert_V\]

Then this structure becomes a sheaf of rings \(\mathcal{C}^\infty_M\) defined on \(M\). ([Topology] §Sheaves, ⁋Definition 1) The stalk of \(\mathcal{C}^\infty\) at any \(p\in M\) is written \(\mathcal{C}^\infty_{M,p}\), or \(\mathcal{C}^\infty_p\) when there is no confusion. ([Topology] §Presheaves, ⁋Definition 9)

Proposition 1 For any manifold \(M\) and a point \(p\in M\), \(\mathcal{C}^\infty_p\) has the structure of an \(\mathbb{R}\)-algebra.

Proof

To show this, it suffices to define the operations on \(\mathcal{C}^\infty_p\). Choose two elements \(\mathbf{f},\mathbf{g}\) of \(\mathcal{C}^\infty_p\). Then there exist suitable open neighborhoods \(U,V\) of \(p\) such that we may regard \(\mathbf{f}\) and \(\mathbf{g}\) as the germs of \((f,U)\) and \((g,V)\), respectively. Now define \(\mathbf{f}+\mathbf{g}\) as the equivalence class of the function

\[(f\vert_{U\cap V}+g\vert_{U\cap V}, U\cap V)\]

In other words, to compute the sum of two germs \(\mathbf{f}\) and \(\mathbf{g}\), we find an open neighborhood of \(p\) on which both functions \(f,g\) are defined, and then compute the sum of \(f\) and \(g\) on this neighborhood. Of course, it is easy to see that this definition does not depend on the choice of representative.

Similarly, one can define multiplication of functions and scalar multiplication.

In fact, for scalar multiplication defined on \(\mathcal{C}^\infty_p\), since this can simply be thought of as multiplication by a constant function, we may regard \(\mathcal{C}^\infty_p\) as a ring rather than an algebra. Thus \((M,\mathcal{C}^\infty_M)\) becomes a ringed space. The next proposition shows that this space is a locally ringed space.

Proposition 2 The ring \(\mathcal{C}^\infty_p\) is a local ring, and its maximal ideal is given by the following expression:

\[\mathfrak{m}_p=\{\mathbf{f}\in \mathcal{C}^\infty_p\mid \mathbf{f}(p)=0\}\]

Proof

First, the condition \(\mathbf{f}(p)=0\) is well-defined. This is because functions belonging to \(\mathbf{f}\) must all have the same value at the point \(p\). It is not difficult to see that \(\mathfrak{m}_p\) is indeed an ideal.

In addition, that \(\mathfrak{m}_p\) is maximal is obtained from the fact that when the evaluation map \(\ev_p:\mathcal{C}^\infty_p\rightarrow\mathbb{R}\) is defined by \(\mathbf{f}\mapsto\mathbf{f}(p)\), the following diagram

\[0\rightarrow \mathfrak{m}_p\rightarrow \mathcal{C}^\infty_p\overset{\ev_p}{\rightarrow}\mathbb{R}\rightarrow 0\]

is exact. This is because \(\mathcal{C}^\infty_p/\mathfrak{m}_p\) is the field \(\mathbb{R}\). Finally, if \(\mathbf{f}\notin\mathfrak{m}_p\), then \(\mathbf{f}(p)\neq 0\), so by continuity a function \(f\) belonging to \(\mathbf{f}\) is not zero on a suitable open neighborhood of \(p\), and since \(1/f\) is a \(C^\infty\) function on that neighborhood, \(\mathbf{f}\) is a unit of \(\mathcal{C}^\infty_p\). Therefore, \(\mathfrak{m}_p\) is the unique maximal ideal of \(\mathcal{C}^\infty_p\).

Tangent Vectors

To summarize the preceding content without the language of sheaves, it is as follows.

We decided to define a tangent vector at the point \(p\) as a directional derivative at the point \(p\). This directional derivative is of course well-defined for any function differentiable on all of \(M\), but since differentiability is essentially a local property, in fact at the point \(p\), the directional derivative of a function defined only on a suitable open neighborhood \(U\) can also be defined.¹

Moreover, if two functions \(f,g\) define, around the point \(p\), the same function on some open neighborhood \(U\), then their derivatives at the point \(p\) also coincide, so when dealing with directional derivatives they may be treated as identical. Then the objects of our interest are no longer functions but equivalence classes of functions, which explicitly is as follows:

\[\mathcal{C}^\infty_p=\{(f,U)\mid p\in U\text{ open}, f\in C^\infty(U)\}\big/{\sim},\qquad (f,U)\sim (g,V)\iff f\vert_W=g\vert_W\text{ for some open $W\subseteq U\cap V$ with $p\in W$}\]

Let us write the equivalence class of \(f\) as \(\mathbf{f}\). Then, as in \(\mathbf{f}\mathbf{g},\mathbf{f}+\mathbf{g}\), scalar multiplication and addition, and furthermore multiplication, are well defined on \(\mathcal{C}^\infty_p\); this is the content of Proposition 1.

Now, a tangent vector is a directional derivative that assigns a real value to each element of \(\mathcal{C}^\infty_p\). Here, the derivative is defined as a linear map satisfying the Leibniz rule.

Definition 3 Consider a manifold \(M\) and a point \(p\in M\). Satisfying the following Leibniz rule

\[v(\mathbf{f}\mathbf{g})=\mathbf{f}(p)v(\mathbf{g})+\mathbf{g}(p)v(\mathbf{f})\]

an \(\mathbb{R}\)-linear map \(v:\mathcal{C}^\infty_p\rightarrow\mathbb{R}\) is called a tangent vector at the point \(p\) of \(M\). The collection of tangent vectors at the point \(p\) of \(M\) is called the tangent space at the point \(p\) of \(M\), and is written \(T_pM\).

The next proposition is a fact that can already be easily guessed from its name.

Proposition 4 The \(T_pM\) defined in Definition 3 is an \(\mathbb{R}\)-vector space.

Proof

Since both \(\mathcal{C}^\infty_p\) and \(\mathbb{R}\) are \(\mathbb{R}\)-vector spaces, \(\Hom_\mathbb{R}(\mathcal{C}^\infty_p,\mathbb{R})\) is also an \(\mathbb{R}\)-vector space. Therefore, to show that the tangent space \(T_pM\) is indeed an \(\mathbb{R}\)-vector space as its name suggests, it suffices to show that \(T_pM\) is closed under addition and scalar multiplication. For example, since \(v+w\) is the linear map defined by

\[(v+w)(\mathbf{f})=v(\mathbf{f})+w(\mathbf{f})\]

to show that \(T_pM\) is closed under addition, we compute

\[\begin{aligned}(v+w)(\mathbf{fg})&=v(\mathbf{fg})+w(\mathbf{fg})=\mathbf{f}(p)v(\mathbf{g})+\mathbf{g}(p)v(\mathbf{f})+\mathbf{f}(p)w(\mathbf{g})+\mathbf{g}(p)w(\mathbf{f})\\ &=\mathbf{f}(p)(v+w)(\mathbf{g})+\mathbf{g}(p)(v+w)(\mathbf{f})\end{aligned}\]

and confirm that \(v+w\) is also an element of \(T_pM\). In the case of scalar multiplication as well, by computing \((cv)(\mathbf{fg})=cv(\mathbf{fg})=\mathbf{f}(p)(cv)(\mathbf{g})+\mathbf{g}(p)(cv)(\mathbf{f})\) we obtain \(cv\in T_pM\).

Moreover, for any tangent vector \(v\) and the constant function with value \(c\) at every point, \(\mathbf{c}\), \(v(\mathbf{c})=0\) always holds. If \(\mathbf{1}\) is the constant function with value \(1\), then since \(\mathbf{c}=c\cdot\mathbf{1}\) it suffices to show only the following:

Lemma 5 For any tangent vector \(v\), we have \(v(\mathbf{1})=0\).

Proof

\[v(\mathbf{1})=v(\mathbf{1}\cdot\mathbf{1})=\mathbf{1}(p)v(\mathbf{1})+\mathbf{1}(p)v(\mathbf{1})=v(\mathbf{1})+v(\mathbf{1})=2v(\mathbf{1}).\]

However, we still do not know what kind of space \(T_pM\) is. In particular, we do not yet know the dimension of \(T_pM\). We will examine this in the next post.

---

References

[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012

---

1. Even for a function defined only on \(U\), via a partition of unity, one can choose a function, agreeing with it on a smaller open neighborhood of the point \(p\), that is defined on all of \(M\) and is \(C^\infty\), so in the present situation we do not need to distinguish the two. ↩