Loose ends of smooth manifolds

 Preliminary exam prep

Here we round up some theorems that have escaped previous roundings-up. Let $X,Y$ be smooth manifolds and $f:X\to Y$ a smooth map.

Theorem: (Inverse function theorem) If $df_p$ is invertible for some $p\in M$, then there exist $U\owns p$ and $V\owns f(p)$ connected such that $f|_U:U\to V$ is a diffeomorphism.

Corollary: (Stack of records theorem) If $\dim(X)=\dim(Y)$, then every regular value $y\in Y$ has a neighborhood $V\owns y$ such that $f^{-1}(Y)=U_1\sqcup \cdots \sqcup U_k$, where $f|_{U_i}:U_i\to V$ is a diffeomorphism.

Proof: Since $y\in Y$ is a regular value, $df_x$ is surjective for all $x\in f^{-1}(y)$. Since $\dim(X)=\dim(Y)$ and $df_x$ is linear, $df_x$ is an isomorphism, hence invertible. By the inverse function theorem, there exist $U\owns x$ and $V\owns y$ connected such that $f|_U:U\to V$ is a diffeomorphism. Before we actually apply this, we need to show that $f^{-1}(y)$ is a finite set.

First we note that by the preimage theorem, since $y$ is a regular value, $f^{-1}(y)$ is a submanifold of $X$ of dimension $\dim(X)-\dim(Y)=0$. Next, if $f^{-1}(y) = \{x_i\}$ were infinite, since $X$ is compact, there would be some limit point $p\in X$ of $\{x_i\}$. But then by continuity,
\[
y = \lim_{i\to \infty}\left[f(x_i)\right] = f\left(\lim_{i\to\infty}\left[x_i\right]\right) = f(p),\]so $p\in f^{-1}(y)$. But then either $p$ cannot be separated from other elements of $f^{-1}(y)$, meaning $f^{-1}(y)$ is not a manifold, or the sequence $\{x_i\}$ is finite in length. Hence $f^{-1}(y) = \{x_1,\dots,x_k\}$. Let $U_i\owns x_i$ and $V_i\owns y$ be the sets asserted to exist by the inverse function theorem (the $U_i$ may be assumed to be disjoint without loss of generality). Let $V = \bigcap_{i=1}^k V_i$ and $U_i' = f^{-1}(V)\cap U_i$, for which we still have $f|_{U_i'}:U_i'\to V$ a diffeomorphism. $\square$

Theorem: (Classification of manifolds) Up to diffeomorphism,

• the only 0-dimensional manifolds are collections of points,
• the only 1-dimensional manifolds are $S^1$ and $\R$, and
• the only 2-dimensional compact manifolds are $S^2\# (T^2)^{\#n}$ or $S^2\#(\R\P^2)^{\#n}$, for any $n\geqslant 0$.

Compact 2-manifolds are homeomorphic iff they are both (non)-orientable and have the same Euler characteristic. Note that
\[
\chi\left(S^2\# (T^2)^{\#n}\right) = 2-2n,
\hspace{1cm}
\chi\left(S^2\#(\R\P^2)^{\#n}\right) = 2-n.
\]
These surfaces are called orientable (on the left) and non-orientable (on the right) surfaces of genus $n$.

Theorem: (Stokes' theorem) For $X$ oriented and $\omega\in \Omega^{n-1}_X$, $\int_X d\omega = \int_{\dy X} \omega$.

Proposition: The tangent bundle $TX$ is always orientable.

Proof: Let $U,V\subset X$ with $\vp:U\to \R^n$ and $\psi:V\to \R^n$ trivializing maps, and $\psi\circ \vp^{-1}:\R^n\to \R^n$ the transition function. To show that $TX$ is always orientable, we need to show the Jacobian of the induced transition function (determinant of the derivative) on $TX$ is always non-negative. On $TU$ and $TV$, we have trivializing maps $(\varphi,d\varphi)$ and $(\psi,d\psi)$, giving a transition function
\[
(\psi\circ \varphi^{-1}, d\psi \circ d\varphi^{-1}) = (\psi\circ \varphi^{-1}, d(\psi \circ \varphi^{-1})). 
\]
The Jacobian of this is
\[
\det(d(\psi\circ \varphi^{-1}, d(\psi \circ \varphi^{-1}))) = \det(d(\psi\circ \varphi^{-1}), d(\psi \circ \varphi^{-1}))) = \det(d(\psi\circ \varphi^{-1}))\cdot \det(d(\psi \circ \varphi^{-1})) \>0,
\]
and since $d(\psi\circ \varphi^{-1})\neq 0$ (as $\psi \circ \varphi^{-1}$ is a diffeomorphism, its derivative is an isomorphism), the result is always positive. $\square$

References: Lee (Introduction to smooth manifolds, Chapter 4), Guillemin and Pollack (Differential topology, Chapter 1

Explicit pushforwards and pullbacks

 Preliminary exam prep

Here we consider a map $f:M\to N$ between manifolds of dimension $m$ and $n$, respectively, and the maps that it induces. Let $p\in M$ with $x_1,\dots,x_m$ a local chart for $U\owns p$ and $y_1,\dots,y_n$ a local chart for $V\owns f(p)$. Induced from $f$ are the differential (or pushforward) $df$ and the pullback $df^*$, which are duals of each other:
\[
\begin{array}{r c l}
df_p\ :\ T_p M & \to & T_{f(p)}N \\[10pt]
df\ :\ TM & \to & TN \\
\alpha & \mapsto & (\beta\mapsto \alpha(\beta\circ f)) \\[10pt]\\\\
\end{array}
\hspace{1cm}
\begin{array}{r c l}
df^*_p\ :\ T_{f(p)}^* N & \to & T_p^*M\\[10pt]
df^*\ :\ T^*N & \to & T^*M \\
\omega & \mapsto & \omega\circ f\\[10pt]
\bigwedge ^k T^*N & \to & \bigwedge^k T^*M\\
\omega\ dy_1\wedge\cdots \wedge dy_k & \mapsto & (\omega \circ f)\ d(y_1\circ f)\wedge \cdots \wedge d(y_k\circ f)
\end{array}
\]
These maps may be described by the diagram below.

Example: For example, consider the map $f:\R^3\to \R^3$ given by $f(x,y,z) = (x-y,3z^2,xz+yz)$, with the image having coordinates $(u,v,w)$. With elements
\[
2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\in TM,
\hspace{2cm}
2uv+\sqrt w-5\in C^\infty(N),
\hspace{2cm}
\cos(uv)\in T^*N,
\]
we have
\begin{align*}
df_p\left(2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\right)(2uv+\sqrt w-5) & = \left(2x\frac\dy{\dy x} - 5z\frac\dy{\dy y}\right)\left(6(x-y)z^2+\sqrt{xz+yz}-5\right)(p),\\
df_p^*\left( \cos(uv)\right) & = \cos((x-y)3z^2), \\
\left(\textstyle\bigwedge^2 df_p^*\right)(\cos(uv)du\wedge dw) & = \cos((x-y)3z^2)d(3z^2)\wedge d(xz+yz) \\
& = \cos((x-y)3z^2)\left(-6z^2\ dx\wedge dz -6z^2\ dy \wedge dz\right).
\end{align*}

The tangent space and differentials

 Preliminary exam prep

Let $M,N$ be smooth $n$-manifolds. Here we discuss different definitions of the tangent space and differentials, or pushforwards, of smooth maps $f:M\to N$.

Derivations (Lee)

Definition: A derivation of $M$ at $p\in M$ is a linear map $v:C^\infty(M)\to \R$ such that for all $f,g\in C^\infty(M)$,
\[
v(fg) = f(p)v(g) + g(p)v(f).
\]
The tangent space $T_pM$ to $M$ at $p$ is the set of all derivations of $M$ at $p$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential $dF_p:T_pM\to T_{f(p)}N$, which, for $v\in T_pM$ and $f\in C^\infty(N)$ acts as
\[
dF_p(v)(f) = v(f\circ F)\in \R.
\]

Dual of cotangent (Hitchin)

Definition: Let $Z_p\subset C^\infty(M)$ be the functions whose derivative vanishes at $p\in M$. The cotangent space $T_p^*M$ to $M$ at $P$ is the quotient space $C^\infty(M)/Z_p$. The tangent space to $M$ at $P$ is the dual of the cotangent space $T_pM = (T_p^*M)^* = \Hom(T_p^*M,\R)$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential
\[
\begin{array}{r c l}
dF_p\ :\ T_pM & \to & T_{F(p)}N, \\
\left(f:C^\infty(M)/Z_p \to \R\right) & \mapsto & \left(\begin{array}{r c l}
g\ :\ C^\infty(N)/Z_{F(p)} & \to & \R, \\ h & \mapsto & f(h\circ F).
\end{array}\right)
\end{array}
\]
This definition makes clear the relation to the first approach. Since $h\not\in Z_{F(p)}$, the derivative of $h$ does not vanish at $F(p)$. Hence the derivative of $h\circ F$ at $p$, which is the derivative of $h$ at $F(p)$ multiplied by the derivative of $F$ at $p$, does not a priori vanish at $p$.

Derivative of chart map (Guillemin and Pollack)

Definition: Let $f:\R^n\to \R^m$ be a smooth map. Then the derivative of $f$ at $x\in \R^n$ in the direction $y\in \R^n$ is defined as
\[
df_x(y) = \lim_{h\to 0}\left[\frac{f(x+yh)-f(x)}h\right].
\]
Given $x\in M$ and charts $\varphi:\R^n\to M\subset \R^m$, the tangent space to $M$ at $p$ is the image $T_pM = d\varphi_0(\R^n)$, where we assume $\varphi(0)=p$.

Given a smooth map $F:M\to N$ and charts $\varphi:\R^n\to M$, $\psi:\R^n\to N$, with $\varphi(0)=p$ and $\psi(0)=F(p)$, define the differential $dF_p:T_pM\to T_{F(p)}N$ via the diagrams below.

Here $h = \psi^{-1}\circ F\circ \varphi$, so $dh_0$ is well-defined. Hence $dF_p = d\psi_0\circ dh_0\circ d\varphi_0^{-1}$ is also well-defined.

Sometimes the differential is referred to as the pushforward, in which case it is denoted by $(F_*)_p$.

References: Lee (Introduction to Smooth Manifolds, Chapter 3), Hitchin (Differentiable manifolds, Chapter 3.2), Guillemin and Pollack (Differential topology, Chapter 1.2)