Differentials

We previously defined, for a manifold \(M\), the tangent space \(T_pM\), showed that their dimension equals that of the original manifold \(M\), and also defined a natural basis of \(T_pM\). In this post, we define functions between two manifolds and examine how they behave on tangent spaces.

Functions between Smooth Manifolds

Definition 1 Let two manifolds \(M,N\) be given. A function \(F:M\rightarrow N\) is said, at a point \(p\in M\), to be \(C^\infty\) if there exist, containing \(p\), a suitable coordinate system \((U,\varphi)\) and, with \(F(U)\subseteq V\), a suitable coordinate system \((V,\psi)\) such that \(\psi\circ F\circ\varphi^{-1}\) is \(C^\infty\).

[Lee], p.34. Fig. 2.2

If \(F\) is \(C^\infty\) at every point, we simply call it a \(C^\infty\) function.

As when we previously defined a function from a manifold to \(\mathbb{R}\) to be \(C^\infty\), one should show that this definition is also independent of the choice of coordinate system; but since this is basically identical to what was proved after §Smooth Manifolds, ⁋Definition 2, we omit it.

Proposition 2 Let two manifolds \(M,N\) be given. If \(F:M\rightarrow N\) is, at a point \(p\in M\), \(C^\infty\), then this function is continuous at \(p\).

Proof

Assume the situation of Definition 1. Then first, the function \(\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)\) between Euclidean spaces is \(C^\infty\). Since this function is differentiable, it is naturally continuous. However, \(\varphi\) and \(\psi\) are both homeomorphisms, so

\[F=\psi^{-1}\circ(\psi\circ F\circ\varphi^{-1})\circ\varphi\]

is continuous as a composition of continuous functions.

Example 3 \(\id_M:M\rightarrow M\) is obviously a \(C^\infty\) function. More generally, if any open subset \(U\subseteq M\) is given the open submanifold structure (§Examples of Differentiable Manifolds, ⁋Definition 3), the inclusion map \(U\hookrightarrow M\) is a \(C^\infty\) function.

Example 4 For any two manifolds \(M,N\), the constant function sending every point of \(M\), \(p\in M\), to a fixed point \(q\in N\) is \(C^\infty\).

We have now defined manifolds, our objects of study, as well as functions between manifolds. It is not difficult to verify that the following proposition holds.

Proposition 5 For three manifolds \(M,N,P\), if both \(F:M\rightarrow N\) and \(G:N\rightarrow P\) are \(C^\infty\), then their composition \(G\circ F\) is also \(C^\infty\).

It is then clear that isomorphisms between manifolds should be defined as follows.

Definition 6 If for two manifolds \(M,N\), there exist \(C^\infty\) functions \(F:M\rightarrow N\) and \(G:N\rightarrow M\) respectively such that \(G\circ F=\id_M\) and \(F\circ G=\id_N\), then each of \(F\) and \(G\) is called a diffeomorphism, and \(M\) and \(N\) are said to be diffeomorphic.

Thus, manifolds and smooth functions form the category \(\Man\).

Remark One can endow the same topological manifold \(M\) with smooth structures that are diffeomorphic but not identical. Let us define two smooth structures \(\mathcal{A}_1\), \(\mathcal{A}_2\) by the single charts \((\mathbb{R},\id_\mathbb{R})\), \((\mathbb{R}, x\mapsto x^{3})\), respectively. Then \(\mathcal{A}_1\) and \(\mathcal{A}_2\) define different smooth structures. (§Smooth Manifolds, ⁋Example 4)
For convenience, let us denote \((M,\mathcal{A}_1)\) by \(M_1\) and \((M,\mathcal{A}_2)\) by \(M_2\), and let \(\varphi=\id_\mathbb{R}\) and \(\psi=(x\mapsto x^3)\).

These two manifolds \(M_1, M_2\) are diffeomorphic to each other. From \(M_1\) to \(M_2\), define a function \(F\) by \(x\mapsto x^{1/3}\). Then obviously \(F^{-1}\) is defined by \(y\mapsto y^3\). By definition, \(F\) is \(C^\infty\). This is because, for any point \(p\in M_1\), if we take on \(M_1\) and \(M_2\) the two coordinate systems \((\mathbb{R},\varphi)\) and \((\mathbb{R},\psi)\) defined respectively, it is obvious that \(p\in\mathbb{R}\) and \(F(\mathbb{R})\subseteq\mathbb{R}\) are satisfied, and furthermore, since they satisfy

\[(\psi\circ F\circ \varphi^{-1})(t)=t\]

\(\psi\circ F\circ\varphi^{-1}\) is \(C^\infty\).
Not only that, \(F^{-1}\) is also \(C^\infty\); this is because, similarly, for any point \(q\in M_2\), if we take the same coordinate systems as above, \(q\in\mathbb{R}\) and \(F^{-1}(\mathbb{R})\subseteq\mathbb{R}\) hold, and also

\[(\varphi\circ F^{-1}\circ \psi^{-1})(s)=s\]

holds.

Differentials

A manifold is fundamentally a space where differentiation can be performed; hence, to understand a function between manifolds, we must know how this function transforms derivations, that is, elements of the tangent space.

Suppose a \(C^\infty\) function \(F:M\rightarrow N\) between two manifolds is given, and fix an arbitrary point \(p\in M\). The function \(F\) naturally induces, via the formula

\[g\mapsto g\circ F\]

the function \(F^\ast:\mathcal{C}_{N,F(p)}^\infty\rightarrow \mathcal{C}_{M,p}^\infty\) defined by it. Moreover, for any \(f,g\in \mathcal{C}_{N,F(p)}^\infty\) and real number \(\alpha\in\mathbb{R}\),

\[F^\ast(f+g)=(f+g)\circ F=f\circ F+g\circ F=F^\ast(f)+F^\ast(g),\quad F^\ast(\alpha f)=(\alpha f)\circ F=\alpha(f\circ F)=\alpha F^\ast(f)\]

holds, so \(F^\ast\) is a linear map between two \(\mathbb{R}\)-vector spaces.

Meanwhile, since \(T_pM\) and \(T_{F(p)}N\) are the elements among the linear maps from \(\mathcal{C}^\infty_{M,p}\) and \(\mathcal{C}^\infty_{N,F(p)}\) to \(\mathbb{R}\) that satisfy the Leibniz rule, they are subspaces of the respective dual spaces \((\mathcal{C}^\infty_{M,p})^\ast\) and \((\mathcal{C}^\infty_{N,F(p)})^\ast\). Therefore, for the linear map \(F^\ast:\mathcal{C}^\infty_{N,F(p)}\rightarrow \mathcal{C}^\infty_{M,p}\) obtained above, we can consider its dual map \((F^\ast)^\ast:(\mathcal{C}^\infty_{M,p})^\ast\rightarrow(\mathcal{C}^\infty_{N,F(p)})^\ast\).

Explicitly, this is the function defined for any linear map \(L\in (\mathcal{C}^\infty_{M,p})^\ast\) by

\[(F^\ast)^\ast(L)=L\circ F^\ast\]

Now, restricting this definition to \(T_pM\) yields the desired definition.

Before that, to restate the above discussion from the viewpoint of the elements of the vector spaces, \((F^\ast)^\ast\vert_{T_pM}\) sends any \(v\in T_pM\) to \(v\circ F^\ast\in (\mathcal{C}^\infty_{N,F(p)})^\ast\). Meanwhile, since \(v\circ F^\ast\) is an element of \((\mathcal{C}^\infty_{N,F(p)})^\ast\), it is defined through how it acts on an arbitrary \(g\in \mathcal{C}^\infty_{N,F(p)}\), which is defined by

\[(v\circ F^\ast)(g)=v(F^\ast(g))=v(g\circ F)\]

Moreover, \(v\circ F^\ast\) defined in this way actually belongs to \(T_{F(p)}N\). That is, it satisfies the Leibniz rule. This is obtained from the following formula:

\[\begin{aligned}(v\circ F^\ast)(fg)&=v(F^\ast(fg))=v((f\circ F)(g\circ F))\\ &=(f\circ F)(p)v(g\circ F)+(g\circ F)(p) v(f\circ F)\\ &=f(F(p))(v\circ F^\ast)(g)+g(F(p))(v\circ F^\ast)(f)\end{aligned}\]

Summarizing the discussion so far yields the following.

Definition 7 Let \(F:M\rightarrow N\) be a \(C^\infty\) function between two manifolds. For any \(p\in M\), the differential of \(F\) at the point \(p\), \(\dd{F_p}:T_pM\rightarrow T_{F(p)}N\), is the linear map defined for any \(v\in T_pM\) and any \(g\in \mathcal{C}^\infty_{N,F(p)}\) by

\[(\dd{F_p}(v))g=v(g\circ F)\]

Several consequences are immediate from the definition. First, for \(\id_M:M\rightarrow M\), \(\dd{(\id_M)_p}\) is always the identity from \(T_pM\) to \(T_pM\), namely \(\id_{T_pM}\). This is clear from the formula in Definition 7. Also, for three manifolds \(M,N,P\), if \(F:M\rightarrow N\) and \(G:N\rightarrow P\) are \(C^\infty\), then the formula

\[\dd{(G\circ F)_p}=(\dd{G_{F(p)}})\circ (\dd{F_p})\]

holds. This is obvious either from the fact that the pullback used to define the differential preserves composition, or likewise by directly substituting \(G\circ F\) into the formula of Definition 7. From this one can show, among other things, that for a diffeomorphism \(F\), \(\dd{F_p}\) is always an isomorphism of vector spaces.

However, there are very many \(C^\infty\) functions whose differential is an isomorphism but which are not diffeomorphisms.

Proposition 8 For a manifold \(M\) and \(M\)’s open submanifold \(U\), the inclusion map \(\iota:U\hookrightarrow M\) induces an isomorphism between tangent spaces for every \(p\in U\). That is, \(\dd{\iota_p}\) is always an isomorphism.

Proof

Since germs are determined on arbitrarily small neighborhoods of a point, the map \(\iota^\ast:\mathcal{C}^\infty_{M,\iota(p)}\rightarrow\mathcal{C}^\infty_{U,p}\) given by restriction is an isomorphism of \(\mathbb{R}\)-algebras, and therefore its dual is also an isomorphism and preserves the Leibniz rule in both directions, so \(\dd{\iota_p}:T_pU\rightarrow T_{\iota(p)}M\) is an isomorphism. In fact, from the outset there is no harm in regarding the two vector spaces as identical.

Bases of Tangent Spaces and Differentials

If we view \(\mathbb{R}^m\) as an \(m\)-dimensional manifold, we can see that the way we defined tangent vectors is exactly identical to directional vectors in \(\mathbb{R}^m\). In this case, for any \(p\in\mathbb{R}^m\), based at the point \(p\), the standard vectors of \(\mathbb{R}^m\), \(m\) in number, define directional derivatives in their respective directions, and we agreed to write these as

\[\frac{\partial}{\partial r^1}\bigg\vert_p,\cdots,\frac{\partial}{\partial r^m}\bigg\vert_p\]

In the case of a general manifold, after choosing a coordinate system containing \(p\in M\), \((U,\varphi)\), we expressed tangent vectors using \(\varphi\)’s component functions \(x^1,\ldots, x^m\) as

\[\frac{\partial}{\partial x^1}\bigg\vert_p,\cdots,\frac{\partial}{\partial x^m}\bigg\vert_p\]

Here, for any \(f\in \mathcal{C}^\infty_{M,p}\),

\[\frac{\partial}{\partial x^i}\bigg\vert_pf=\frac{\partial}{\partial r^i}\bigg\vert_{\varphi(p)} (f\circ\varphi^{-1})\]

holds. However, keeping Definition 7 in mind and examining this formula again, we see that it has the same form as the differential of \(\varphi^{-1}:\varphi(U)\rightarrow U\).¹ That is, the basis of the tangent space is nothing other than transferring, from \(\mathbb{R}^m\)’s tangent space \(T_{\varphi(p)}\mathbb{R}^m\), the \(m\) basis vectors via the differential \(\dd{\varphi}^{-1}_{\varphi(p)}\).

To view this from a more linear-algebraic perspective, if we let \(\mathcal{B}\) be the standard basis of \(\mathbb{R}^m\) and \(\mathcal{C}\) be the basis consisting of the \(\partial/\partial x^i\) for \(T_pM\), then from \((T_{\varphi(p)}\mathbb{R}^m, \mathcal{B})\) to \((T_pM, \mathcal{C})\), the matrix representation of the linear map \(\dd{\varphi}^{-1}_{\varphi(p)}\) is precisely the identity matrix.

More generally, let \(M,N\) be manifolds of dimensions \(m,n\) respectively, and let \(F:M\rightarrow N\) be any \(C^\infty\) function. Then for a fixed \(p\in M\), there exist a coordinate system containing \(p\), \((U,\varphi)\), and a coordinate system containing \(F(U)\), \((V,\psi)\), such that \(\psi\circ F\circ\varphi^{-1}\) is \(C^\infty\). Now let \(\varphi=(x^i)_{i=1}^{m}\) and \(\psi=(y^j)_{j=1}^n\). Then likewise, the bases of the tangent spaces \(T_pM\) and \(T_{F(p)}N\) are given respectively by

\[\frac{\partial}{\partial x^1}\bigg\vert_p,\cdots,\frac{\partial}{\partial x^m}\bigg\vert_p,\quad\text{and}\quad\frac{\partial}{\partial y^1}\bigg\vert_{F(p)},\cdots\frac{\partial}{\partial y^n}\bigg\vert_{F(p)}\]

Let us now represent \(\dd{F_p}\) as a matrix with respect to these. To do this, it suffices to express the vector to which each \(\partial/\partial x^i\) is sent via \(\dd{F_p}\) as a linear combination of the \(\partial/\partial y^j\). That is, in

\[\dd{F_p}\left(\frac{\partial}{\partial x^i}\bigg\vert_p\right)=a_{1i}\frac{\partial}{\partial y^1}\bigg\vert_{F(p)}+\cdots+a_{ni}\frac{\partial}{\partial y^n}\bigg\vert_{F(p)}\]

it suffices to find each of the coefficients \(a_{ji}\). However, if we denote the maximal ideal of \(\mathcal{C}^\infty_{N,F(p)}\) by \(\mathfrak{n}\), since as seen in §Cotangent Space, ⁋Definition 3 anyway, the \(\partial/\partial y^j\) are the dual basis to the elements of \(\mathfrak{n}/\mathfrak{n}^2\), \(y^j+\mathfrak{n}^2\), for this it suffices to apply both sides to the function \(y^j\).² That is, from

\[\dd{F_p}\left(\frac{\partial}{\partial x^i}\bigg\vert_p\right)y^j=a_{1i}\frac{\partial}{\partial y^1}\bigg\vert_{F(p)}y^j+\cdots+a_{ji}\frac{\partial}{\partial y^j}\bigg\vert_{F(p)}y^j+\cdots+a_{ni}\frac{\partial}{\partial y^n}\bigg\vert_{F(p)}y^j\]

by the definition of the dual basis only \(a_{ji}\) remains on the right-hand side, so

\[\dd{F_p}\left(\frac{\partial}{\partial x^i}\bigg\vert_p\right)y^j=a_{ji}\]

and from this we see that the matrix representation of \(\dd{F_p}\) with respect to the two bases \(\partial/\partial x^i\) and \(\partial/\partial y^j\) is the matrix

\[\begin{pmatrix}\partial(y^1\circ F)/\partial x^1&\partial(y^1\circ F)/\partial x^2&\cdots&\partial(y^1\circ F)/\partial x^m\\\partial(y^2\circ F)/\partial x^1&\partial(y^2\circ F)/\partial x^2&\cdots&\partial(y^2\circ F)/\partial x^m\\\vdots&\vdots&\ddots&\vdots\\\partial(y^n\circ F)/\partial x^1&\partial(y^n\circ F)/\partial x^2&\cdots&\partial(y^n\circ F)/\partial x^m\end{pmatrix}\]

That is, this is nothing but the Jacobian of the function \(\psi\circ F\circ\varphi^{-1}\) between Euclidean spaces.

In particular, if \(M=N\) and \(F=\id_M\) but we choose different coordinate systems \((U, \varphi)\) and \((V,\psi)\), this will be the Jacobian matrix of the transition map \(\psi\circ\varphi^{-1}\).

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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

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1. Of course, for this \(\varphi^{-1}\) must be \(C^\infty\), but since the differentiable structure on \(U\) is itself just the differentiable structure of \(\varphi(U)\) transferred over in the first place, \(\varphi^{-1}\) is in fact a diffeomorphism. ↩

2. For convenience, we assumed \(\psi(F(p))=0\). ↩