Manifolds, First Look

A manifold is a space that looks like ordinary Euclidean space when viewed in a sufficiently small neighborhood, even if its global shape is curved, periodic, or otherwise not a single copy of $\mathbb R^n$.

The slogan is:

A manifold is locally coordinatized, but globally it may need several coordinate patches.

This first-look page gives the minimum geometric language needed for Berry phase, angular variables, constrained configuration spaces, Bloch spheres, and topological examples. It does not try to replace a differential-geometry course.

Helpful background. Sets, Functions, and Maps explains maps and inverses; the coordinate examples use ordinary multivariable derivatives.

Manifolds in quantum mechanics

Angles provide a useful first example. An angle can label points near any chosen direction, but after a full turn it returns to the same point. Local coordinates describe nearby changes; periodic identifications describe the whole space. The circle, sphere, and torus below show how to keep those two ideas separate.

Local Coordinates

An $n$-dimensional manifold $M$ is a space in which each point $p\in M$ has a neighborhood $U$ that can be assigned coordinates in an open region of $\mathbb R^n$.

A chart is a map

$$\varphi:U\to V\subset\mathbb R^n,$$

where $U$ is an open patch of $M$, $V$ is an open set in Euclidean space, and $\varphi$ is a homeomorphism: both it and its inverse are continuous. We assume the usual Hausdorff and countability conditions, which exclude pathological spaces. If

$$\varphi(p) = (x^1(p),\ldots,x^n(p)),$$

then $x^1,\ldots,x^n$ are local coordinates on that patch.

Local coordinates let one write formulas. The manifold is the coordinate-independent space being described.

Atlases and Coordinate Changes

One chart is often not enough. An atlas is a collection of charts whose domains cover the manifold.

If two charts overlap,

$$\varphi_\alpha:U_\alpha\to V_\alpha, \qquad \varphi_\beta:U_\beta\to V_\beta,$$

then points in the overlap have two coordinate descriptions. The coordinate-change map is

$$\varphi_\beta\circ\varphi_\alpha^{-1}.$$

For a smooth manifold, these coordinate changes are smooth wherever they are defined. This is the mathematical way of saying that different coordinate systems describe the same local geometry without tearing or folding it.

Circle

The unit circle is

$$S^1 = \left\{ (x,y)\in\mathbb R^2: x^2+y^2=1 \right\}.$$

It can be parametrized by an angle:

$$(x,y) = (\cos\theta,\sin\theta).$$

But $\theta$ is not a global one-to-one coordinate because

$$\theta \sim \theta+2\pi n, \qquad n\in\mathbb Z.$$

One can cut the circle at a point and use an angle coordinate on the remaining open arc. A different cut gives another chart. The need for multiple charts is not a pathology; it is the ordinary global structure of $S^1$.

This is the same practical issue that appears whenever phase is defined only modulo $2\pi$.

Sphere

The unit sphere is

$$S^2 = \left\{ (x,y,z)\in\mathbb R^3: x^2+y^2+z^2=1 \right\}.$$

Spherical coordinates write

$$\begin{aligned} x&=\sin\theta\cos\phi,\\ y&=\sin\theta\sin\phi,\\ z&=\cos\theta. \end{aligned}$$

These coordinates are familiar, but they are not a single smooth global chart. At the north and south poles, the azimuthal angle $\phi$ is undefined. The poles are not singular points of the sphere; the coordinates are singular there.

This distinction matters in angular momentum and Berry phase. A formula can look singular because of a chart choice even when the underlying state or direction is regular.

Torus

A two-torus can be represented by two independent angles:

$$T^2 = S^1\times S^1.$$

Coordinates may be written as

$$(\theta,\phi),$$

with the identifications

$$(\theta,\phi) \sim (\theta+2\pi m,\phi+2\pi n), \qquad m,n\in\mathbb Z.$$

Tori appear in periodic boundary conditions, phase variables, Brillouin-zone models, and integrable classical systems. The important feature is not the donut picture; it is the double periodicity.

Parameter Spaces

In Berry-phase problems, one often studies a Hamiltonian

$$H(R),$$

where $R$ denotes external parameters. The parameter space may be a line, a plane with a degeneracy removed, a sphere of field directions, a torus of phases, or a more abstract space of controls.

A path in parameter space is a map

$$R:[0,T]\to M.$$

A closed loop satisfies

$$R(T)=R(0).$$

The Berry phase depends on the loop traced in this space, not merely on a list of local coordinate values. For the physical adiabatic phase, see Berry Phase.

Manifolds versus Vector Spaces

Every finite-dimensional real vector space is a manifold, but not every manifold is a vector space.

For example, $\mathbb R^2$ is both a vector space and a manifold. The sphere $S^2$ is a manifold, but it is not a vector space: adding two points on the sphere as ordinary vectors in $\mathbb R^3$ does not generally give another point on the sphere.

This distinction prevents a common mistake. Local coordinates may look like vectors, but the points of the manifold do not automatically inherit a global addition operation.

Boundaries, Removed Points, and Singularities

Some spaces have boundaries, such as an interval $[0,L]$ or a disk. These are manifolds with boundary, and boundary conditions become part of the physics.

Some spaces are built by removing points. The punctured plane

$$\mathbb R^2\setminus\{0\}$$

is locally two-dimensional everywhere, but loops around the missing point cannot be shrunk to a point. This kind of global structure is central to Aharonov–Bohm physics and winding-number examples.

Removing a point changes the space; it does not make the remaining punctured plane a singular manifold. A true nonmanifold point would fail to have a Euclidean neighborhood, as at the crossing of two intersecting lines. The removed origin is not a point of the punctured plane at all. By contrast, the north pole is an ordinary point of $S^2$ even though spherical coordinates behave badly there.

From coordinates to calculus

Tangent and Cotangent Spaces explains how to describe directions of motion and derivatives at a point. Differential Forms then turns those derivatives into integrands for paths and surfaces. You can return to bundles and connections when a Berry-phase calculation requires them.

Common Mistakes

• Thinking a manifold must be embedded in a higher-dimensional Euclidean space.
• Treating a local coordinate system as a global description.
• Confusing coordinate singularities with physical singularities.
• Assuming every manifold has a natural origin or vector addition.
• Forgetting periodic identifications such as $\theta\sim\theta+2\pi$.
• Treating a parameter space as ordinary Euclidean space after removing degeneracy points.
• Using topology words without specifying which space is being discussed.

Exercises

1. Why is a single angle $\theta$ not a global one-to-one coordinate on $S^1$?

Solution

The parametrization

$$(\cos\theta,\sin\theta)$$

is periodic:

$$(\cos(\theta+2\pi),\sin(\theta+2\pi)) = (\cos\theta,\sin\theta).$$

Thus $\theta$ and $\theta+2\pi$ label the same point. A one-to-one coordinate chart must avoid this redundancy, usually by cutting the circle at a point and using another chart near the cut.

2. In spherical coordinates, why is the north pole not a singular point of the sphere?

Solution

At the north pole, $\theta=0$ and all values of $\phi$ describe the same point. The failure is in the coordinate description: the azimuthal angle is undefined there. The sphere itself is smooth at the north pole, and another chart can describe a neighborhood of that point without the same coordinate singularity.

3. What identifications define a two-torus in angle coordinates?

Solution

A two-torus has two periodic coordinates. Thus

$$(\theta,\phi) \sim (\theta+2\pi m,\phi+2\pi n), \qquad m,n\in\mathbb Z.$$

The two integers record independent windings around the two circle factors.

4. Why is $S^2$ not a vector space, even though it sits inside $\mathbb R^3$?

Solution

The sum of two unit vectors is not generally a unit vector. For example, adding two nearby points on $S^2$ as vectors in $\mathbb R^3$ gives a vector whose length is not $1$. Therefore $S^2$ is not closed under vector addition and is not a vector space.

5. A Hamiltonian depends on two parameters and has a degeneracy at the origin, so the parameter space used for adiabatic loops is $\mathbb R^2\setminus\{0\}$. What global feature should you notice?

Solution

Loops can wind around the missing origin. Such loops cannot always be continuously shrunk to a point without crossing the removed degeneracy. This global feature is exactly the kind of structure that can matter for Berry phases and related holonomy effects.

References

• T. Frankel, The Geometry of Physics, 3rd ed., Cambridge University Press, 2011.
• J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013.
• L. W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011.