MAT257Y1/Unit 13
Unit 13 · Analysis II

Manifolds in $\mathbb{R}^n$

A $k$-manifold is a subset of $\mathbb{R}^n$ that looks locally like $\mathbb{R}^k$. Three definitions — local graph, local parametrisation, regular level set — describe the same thing, and the implicit function theorem is what proves it. Once orientation is added, a $k$-form can be integrated over a $k$-manifold, and the last piece of Stokes' theorem is in place.

By the end you can
Why it matters & where it comes from
Why it exists

Spheres, tori, configuration spaces of linkages and solution sets of equations are not vector spaces, but calculus still needs to run on them. Local flatness is the minimum assumption that makes derivatives and integrals meaningful, and it is exactly what the definition captures.

In the real world

The state space of a robot arm is a torus; the space of rotations is a three-manifold; constraint surfaces in optimisation are level sets. Numerical methods that stay on a manifold — geodesic optimisers, Lie group integrators — are written directly in this language.

Story & open questions

Riemann proposed manifolds in 1854; Whitney proved in 1936 that any abstract manifold embeds in some $\mathbb{R}^n$, which is why the concrete definition used here loses no generality. Classifying manifolds is open in dimension four and solved in every other dimension.

Three equivalent definitions

Definition $k$-manifold

$M\subseteq\mathbb{R}^n$ is a $k$-dimensional $C^r$ manifold if each $p\in M$ has an open neighbourhood $W$ in $\mathbb{R}^n$ for which one (hence all) of the following holds:

Graph. After permuting coordinates, $M\cap W=\{(x,g(x))\}$ for a $C^r$ map $g$ from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^{n-k}$.
Parametrisation. There is an open $U\subseteq\mathbb{R}^k$ and a $C^r$ map $\alpha:U\to\mathbb{R}^n$ that is a homeomorphism onto $M\cap W$ with $D\alpha$ of rank $k$ everywhere.
Level set. There is a $C^r$ map $F:W\to\mathbb{R}^{n-k}$ with $M\cap W=F^{-1}(0)$ and $JF$ of rank $n-k$ throughout.
Proof Why they agree
Graph gives parametrisation: take $\alpha(x)=(x,g(x))$, whose derivative has an identity block and so rank $k$.
Graph gives level set: take $F(x,y)=y-g(x)$, whose Jacobian contains an identity block of size $n-k$.
Level set gives graph: that is exactly the implicit function theorem of Unit 7.
Parametrisation gives graph: $D\alpha$ has rank $k$, so some $k$ rows are independent; the inverse function theorem applied to those $k$ components makes them local coordinates, and the rest become functions of them. ∎

Each of the two big theorems from the middle of the course is used once. This is the payoff for Units 6 and 7.

Example The sphere three ways

$S^2\subseteq\mathbb{R}^3$: as a level set, $F(x)=\|x\|^2-1$ with $\nabla F=2x\ne0$ on $S^2$; as graphs, six hemispherical caps $z=\pm\sqrt{1-x^2-y^2}$ and their permutations; as parametrisations, spherical coordinates on the complement of a meridian. No single chart covers it, and that is typical rather than exceptional.

Example Not a manifold

The cone $z^2=x^2+y^2$ fails at the apex: no neighbourhood of it is a graph over any pair of coordinates in a $C^1$ way. Removing one point repairs it. Similarly the union of the two axes in $\mathbb{R}^2$ fails at the crossing, where the set is locally two lines rather than one.

Tangent spaces and manifolds with boundary

Definition Tangent space

$T_pM$ is the set of $\gamma'(0)$ over $C^1$ curves $\gamma$ in $M$ with $\gamma(0)=p$. It is a $k$-dimensional subspace, computable two ways:

$T_pM=\operatorname{im}D\alpha(u)$ for a parametrisation $\alpha$ with $\alpha(u)=p$;
$T_pM=\ker JF(p)$ for a defining map $F$.

Image of the parametrisation, kernel of the constraint. Which is easier depends on how $M$ was handed to you.

Definition Manifold with boundary

Allow charts modelled on the half space $H^k=\{u\in\mathbb{R}^k:u_k\ge0\}$. Points mapping to $u_k=0$ form the boundary $\partial M$, which is itself a manifold of dimension $k-1$, and it is empty or has no boundary of its own — $\partial(\partial M)=\emptyset$. That fact is the geometric half of $d^2=0$.

Example Boundaries

The closed unit ball in $\mathbb{R}^3$ is a $3$-manifold with boundary $S^2$. The closed upper hemisphere is a $2$-manifold with boundary the equator. $S^2$ itself has no boundary, which is why the manifold boundary and the topological boundary of a set must not be confused: as a subset of $\mathbb{R}^3$, $S^2$ equals its topological boundary.

Orientation and integration

Definition Orientation

An orientation of a $k$-manifold is a continuous choice of orientation of each tangent space — equivalently, an atlas of charts whose transition maps all have positive Jacobian determinant. A manifold admitting one is orientable.

Equivalently again: an orientation is a nowhere-vanishing $k$-form on $M$, called a volume form. Existence of such a form is the cleanest test.

Example The Möbius band

Transport a chosen normal once around the core circle of a Möbius band and it comes back reversed. No continuous choice exists, so the band is not orientable, and no integral of a $2$-form over it can be defined. The Klein bottle is the closed example.

Definition The integral of a form

For an oriented $k$-manifold $M$ and a $k$-form $\omega$ supported inside one orientation-preserving chart $\alpha:U\to M$,

$\displaystyle\int_M\omega=\int_U\alpha^{*}\omega,$

the right side being an ordinary integral of a function over $U\subseteq\mathbb{R}^k$. In general, take a partition of unity $\{\rho_i\}$ subordinate to a cover by charts and set $\int_M\omega=\sum_i\int_M\rho_i\omega$.

Theorem It is well defined

Two orientation-preserving charts differ by a transition map $\varphi$ with $\det J\varphi>0$. Pulling back a top form multiplies by $\det J\varphi$ (Unit 11), and the change of variables formula (Unit 10) divides by $|\det J\varphi|$. Positivity makes the two cancel exactly. Independence of the partition of unity follows from bilinearity and local finiteness. ∎

This single computation is why orientation is a hypothesis and not a convenience: with $\det J\varphi<0$ allowed, the two charts would give answers differing in sign.

Every tool in the course meets here. Charts come from the implicit function theorem, the gluing comes from partitions of unity, the consistency comes from change of variables, and the integrand comes from the exterior algebra. Nothing in the definition of $\int_M\omega$ is new; it is all assembly.
Common pitfalls

• Confusing the manifold boundary with the topological boundary of a subset of $\mathbb{R}^n$. For $S^2$ they are completely different.
• Assuming a parametrisation covers the whole manifold. Compact manifolds usually need several charts.
• Forgetting the rank condition on $D\alpha$. Without it, $t\mapsto(t^2,t^3)$ parametrises a cusp.
• Integrating a $j$-form over a $k$-manifold with $j\ne k$. Only matching degrees make sense.
• Assuming orientability. The Möbius band is the standing counterexample and it is not exotic.

Practice

Original problems written for this guide in the style of Spivak and Munkres exercises. Write a full solution before opening the answer.

Practice 1 · Level set check

Show the torus $(\sqrt{x^2+y^2}-2)^2+z^2=1$ is a $2$-manifold.

Solution
Let $F$ be the left side minus $1$. Away from the $z$-axis, $F$ is smooth and $\nabla F\ne0$ on the level set: the $z$-component is $2z$ and the radial component is $2(\sqrt{x^2+y^2}-2)$, and both vanish only when $\sqrt{x^2+y^2}=2$ and $z=0$, which is not on the surface. Rank $1$ throughout, so it is a $2$-manifold.
Practice 2 · Tangent plane

Find $T_pS^2$ at $p=(0,0,1)$.

Solution
$\ker\nabla F(p)$ with $\nabla F=2p=(0,0,2)$, so the tangent space is the $xy$-plane.
Practice 3 · Rank matters

Why is $t\mapsto(t^2,t^3)$ not a manifold parametrisation?

Solution
Its derivative $(2t,3t^2)$ vanishes at $t=0$, so the rank condition fails there. The image has a cusp at the origin and is not a $1$-manifold.
Practice 4 · Graph of anything smooth

Show the graph of any $C^1$ map $g:\mathbb{R}^k\to\mathbb{R}^{n-k}$ is a $k$-manifold.

Solution
It is globally a graph, which is the first definition with a single chart. Alternatively $F(x,y)=y-g(x)$ has full rank everywhere.
Practice 5 · Orthogonal matrices

Show $O(n)=\{A:A^{T}A=I\}$ is a manifold and find its dimension.

Solution
$F(A)=A^{T}A-I$ maps into the symmetric matrices, of dimension $n(n+1)/2$. Its derivative $H\mapsto A^{T}H+H^{T}A$ is surjective onto the symmetric matrices at each $A\in O(n)$: given symmetric $S$, take $H=\tfrac12AS$. So $O(n)$ has dimension $n^2-n(n+1)/2=n(n-1)/2$.
Practice 6 · Boundary of a boundary

Why is $\partial(\partial M)$ empty?

Solution
Near a boundary point the manifold looks like $H^k$, and $\partial H^k$ is a copy of $\mathbb{R}^{k-1}$, which has no boundary. The claim is local and this is the local picture.
Practice 7 · Integrating over a circle

Compute $\int_{S^1}(-y\,dx+x\,dy)$ with the counterclockwise orientation.

Solution
Parametrise by $\alpha(t)=(\cos t,\sin t)$ on $[0,2\pi]$. The pullback is $(\sin^2t+\cos^2t)\,dt=dt$, so the integral is $2\pi$. This is the angle form from Unit 12, and the answer is why it is not exact.
Practice 8 · Orientability by volume form

Exhibit a nowhere-zero $2$-form on $S^2$.

Solution
Restrict $\omega=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy$ to $S^2$. On a tangent basis it evaluates to the determinant with the outward normal $p$, which is nonzero since $\|p\|=1$. So $S^2$ is orientable.

Expanded topic checklist

Course coverage. This unit is part of the complete MAT257Y1 topic map. It should cover the following items before you move on:

Textbook/source alignment

Spivak defines manifolds through the local diffeomorphism condition and moves fast; Munkres separates the three characterisations and proves the equivalences in detail. Match the syllabus for which definition is primary.

Core theorem map

Implicit and inverse function theorems give the equivalence of the three definitions. Change of variables plus positive transition determinants make $\int_M\omega$ well defined. Unit 14 pairs it with $d$ and the boundary.

Study routine. For every surface you meet, write down a defining $F$, a parametrisation, and the tangent space computed both ways. When the two tangent spaces agree you have understood the equivalence rather than memorised it.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Which theorem

Which theorem proves level sets are locally graphs?

Guide
The implicit function theorem, applied where the Jacobian has full rank.
Extension 2 · Two formulas for one space

Give the tangent space as image and as kernel.

Guide
$\operatorname{im}D\alpha$ for a parametrisation, $\ker JF$ for a defining map.
Extension 3 · Orientation test

What is the cleanest criterion for orientability?

Guide
Existence of a nowhere-vanishing top-degree form on the manifold.
Extension 4 · Why partitions of unity

Why is a partition of unity needed to define $\int_M\omega$?

Guide
Because one chart rarely covers $M$. The partition splits $\omega$ into chart-sized pieces whose integrals add, independently of the splitting chosen.
← Unit 12The exterior derivative Unit 14 →Stokes' theorem