Theory

Topological and smooth manifolds

Topological manifold

A (second-countable, Hausdorff) topological space $X$ is an $n$-dimensional (topological) manifold if every point $p\in X$ has an open neighborhood $U$ which is homeomorphic to ${\mathbb{R}}^n$.

$$\text{“ the local topology of an n-manifold is that of Rn ”}$$

Notes:

• An equivalent definition is that every $p$ has open neighborhood $U$ homeomorphic to the open ball ${\mathbb{B}}_1(0)\in {\mathbb{R}}^n$, since of course ${\mathbb{B}}_1(0)$ is itself homeomorphic to ${\mathbb{R}}^n$.
• It is a theorem that every $n$-manifold can be embedded in some Euclidean space ${\mathbb{R}}^N$, so an equivalent definition states adds the hypothesis that $X\subset {\mathbb{R}}^N$ for some $N$.

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To define smooth manifolds, we must first define differentiable and smooth maps.

Regular points

Suppose $p\in V\subset {\mathbb{R}}^n$ with $V$ open. Let $(x_1,\dots ,x_n)$ give coordinates on ${\mathbb{R}}^n$.

Suppose $f:V\to {\mathbb{R}}^d$ is a function given in coordinates by $f(x_1,\dots ,x_n)=(f_1,\dots ,f_d)$.

The differential matrix of $f$ at $p=(x_1,\dots ,x_n)$ is the Jacobi matrix:

$$[d_{p}f]=\left(\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}(p)& \cdots & \frac{\partial f_1}{\partial x_n}(p)\\ ⋮& \ddots & ⋮\\ \frac{\partial f_d}{\partial x_1}(p)& \cdots & \frac{\partial f_d}{\partial x_n}(p)\end{array}\right)$$

The point $p$ is regular for $f$ when $[df]$ has rank $n$.
(No point can be regular when $d<n$.)

Smooth manifold in ${\mathbb{R}}^d$

A subspace $M\subset {\mathbb{R}}^d$ is a smooth $n$-manifold if every $p\in M$ has an open neighborhood in $M$ (open in the subspace topology) that is the image of a smooth injective function $\phi :U\to {\mathbb{R}}^d$, whose domain is an open set $U\subset {\mathbb{R}}^n$, and whose differential $[d\phi ]$ has maximal rank everywhere on $U$.

Each such function is called a parametrization or a coordinate chart for $M$, and the image of a parametrization is a coordinate patch on $M$.

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A parametrized surface is a bijective and infinitely differentiable (smooth) injective map:

$$\phi :U\to {\mathbb{R}}^3$$

where the parameter domain $U$ is an open subset of ${\mathbb{R}}^2$. Every parametrized surface defines a $2$-manifold in ${\mathbb{R}}^3$.

Using coordinates $(x,y)$ on ${\mathbb{R}}^2$ and $({\phi }_1,{\phi }_2,{\phi }_3)$ for the images of $\phi$ in ${\mathbb{R}}^3$, we have for the differential matrix:

$$[d_{(x,y)}\phi ]=\left(\begin{array}{cc}\frac{\partial {\phi }_1}{\partial x}& \frac{\partial {\phi }_1}{\partial y}\\ \frac{\partial {\phi }_2}{\partial x}& \frac{\partial {\phi }_2}{\partial y}\\ \frac{\partial {\phi }_3}{\partial x}& \frac{\partial {\phi }_3}{\partial y}\end{array}\right)$$

The maximum rank of $[d\phi ]$ is $2$, and $\mathrm{Rank}[d\phi ]=2$ occurs when any of these equivalent conditions are satisfied:

• The columns of $[d\phi ]$ are linearly independent (for two columns: not collinear vectors)
• There are two linearly independent rows in $[d\phi ]$
• There is a $2\times 2$ submatrix of $[d\phi ]$ with nonzero determinant

If the matrix $[d\phi ]$ is considered as a linear map $[d\phi ]:{\mathbb{R}}^2\to {\mathbb{R}}^3$, then the image of $[d\phi ]$ at $p=(x,y)$ is a plane through the origin which is parallel to the tangent plane to the surface at $\phi (x,y)$.

The condition $\mathrm{Rank}[d\phi ]=2$ ensures that small movements in the parameter space, $dx$ and $dy$, can generate two dimensions of movement on the surface. In turn this ensures that locally around $\phi (x,y)$ the surface is homeomorphic to an open disk in ${\mathbb{R}}^2$, and moreover that such a homeomorphism can be constructed as projection to the tangent plane at $\phi (x,y)$ composed with a linear map.

Local coordinates

Smooth map

A map $\phi :U\to V$ for open sets $U\subset {\mathbb{R}}^n$ and $V\subset {\mathbb{R}}^d$ is smooth when all partial derivatives of all orders exist and are continuous.

A map $\phi$ is a diffeomorphism when it is a homeomorphism and both $\phi$ and ${\phi }^{-1}$ are smooth.

Suppose we have $\phi :U\to {\mathbb{R}}^d$ a smooth map for $U\subset {\mathbb{R}}^n$ open, $n\le d$, and we know $\phi$ is injective and $[d_p\phi ]$ has maximal rank for all $p\in U$.

Then the Inverse Function Theorem from multivariable real analysis implies that ${\phi }^{-1}$ is also a smooth map from $\phi (U)$ to $U$.

Therefore the parametrizations (charts) of a smooth manifold are diffeomorphisms.

Local coordinates

Given a coordinate chart $\phi$ for a smooth manifold $M$, the coordinates of the images of the inverse ${\phi }^{-1}$ provide local coordinates on $M$.