Manifold

Prerequisites: Calculus, Linear Algebra.

• Why is it interesting?
• Intuitive
• Concrete
• Abstract
• Examples

Why is it interesting?

Manifolds are especially important in General Relativity. Through massive objects, spacetime is curved and no longer flat. Manifolds are the correct mathematical tool to describe spacetime when it isn't flat.

Intuitive

The surface of the Earth is curved, but a city map drawn on a flat sheet of paper works perfectly well for finding your way around town. No single flat map can show the whole globe without tearing or overlapping somewhere, so an atlas uses many maps, each covering a region, and neighbouring maps overlap so that we know how to continue from one page to the next. A manifold is any space that can be described like this: every small region looks like a piece of ordinary flat space of some fixed dimension, and the whole space is assembled from such regions glued together along their overlaps.

The surface of a sphere and the surface of a doughnut are two-dimensional manifolds. A figure eight is not a manifold, because near the crossing point it does not look like a piece of a line. The dimension is the number of coordinates needed locally: two for a surface, four for spacetime.

The reason this matters in physics is that calculus only needs local information. Derivatives, velocities and differential equations are defined in small regions, and on a manifold every small region looks flat, so we can do calculus there even if the space as a whole is curved or has a complicated shape. This is exactly what general relativity requires: spacetime is curved by matter, but in a small enough region around any event it looks like the flat spacetime of special relativity. The configuration spaces of mechanical systems are manifolds for the same reason, for example the angle of a pendulum lives on a circle.

• What Is A Manifold?

Concrete

The standard sentence is: A Manifold is a set of points that locally "looks the same" as some Euclidean space. Such geometric spaces are particularly nice to investigate, because we can use, locally, the familiar tools from elementary calculus. Happily, such spaces are not only nice to investigate, but also useful in physics to describe nature.

The idea of locality is made precise by the mathematical concept of a neighborhood. To each point of the manifold, a neighborhood must exist that is just like a small piece of Euclidean space.

The idea of "looks like Euclidean space" is made precise by the concept of a diffeomorphism. If there exists a map from each neighborhood onto some piece of Euclidean space that is smooth, one-to-one, onto, and has a smooth inverse (a diffeomorphism), the space in question is a manifold.

This map from some neighborhood $X$ of the manifold onto Euclidean space gives us local coordinates of the points in this neighborhood. The inverse map is called a parametrization of the neighborhood. Therefore another way of thinking about an n-dimensional manifold is that its a set which can be given n independent coordinates ins some neighborhood of any point.

Abstract

A topological manifold of dimension $n$ is a Hausdorff, second-countable topological space $M$ in which every point has an open neighbourhood homeomorphic to an open subset of $\mathbb{R}^n$. A chart is such a homeomorphism $\varphi_\alpha:U_\alpha\to\varphi_\alpha(U_\alpha)\subset\mathbb{R}^n$, and an atlas is a family of charts whose domains cover $M$. Where two domains overlap there is a transition map $$\varphi_\beta\circ\varphi_\alpha^{-1}:\varphi_\alpha(U_\alpha\cap U_\beta)\to\varphi_\beta(U_\alpha\cap U_\beta),$$ a homeomorphism between open subsets of $\mathbb{R}^n$. If all transition maps are $C^\infty$, the atlas is smooth, and a maximal smooth atlas is a smooth structure. A smooth manifold is a topological manifold with a smooth structure. The Hausdorff and second-countability conditions exclude pathological examples such as the line with two origins, and they guarantee partitions of unity, which are used to glue local constructions (metrics, integrals) into global ones.

The point of the smooth structure is that every notion of calculus that is invariant under smooth changes of coordinates can be transferred to $M$. A function $f:M\to\mathbb{R}$ is smooth if $f\circ\varphi_\alpha^{-1}$ is smooth for every chart. The tangent space $T_pM$ can be defined as the space of derivations of smooth functions at $p$; in a chart it has the basis $\partial/\partial x^\mu|_p$, and under a change of coordinates its components transform with the Jacobian of the transition map. The tangent spaces form the tangent bundle $TM$, and its dual and tensor powers give covectors, tensor fields and differential forms. Additional structure is put on top of the manifold: a Riemannian or Lorentzian metric for gravity, a symplectic form for Hamiltonian mechanics, a fibre bundle for gauge fields.

Some standard facts: by the Whitney embedding theorem, every smooth $n$-manifold embeds in $\mathbb{R}^{2n}$, so abstract manifolds and submanifolds of Euclidean space are the same class of objects; the abstract definition simply avoids an embedding that has no physical meaning, such as an ambient space for spacetime. A given topological manifold may admit no smooth structure, or several inequivalent ones: Milnor found smooth structures on the topological 7-sphere that are not diffeomorphic to the standard one, and $\mathbb{R}^n$ has a unique smooth structure up to diffeomorphism for every $n\neq4$, while $\mathbb{R}^4$ has uncountably many.

In general relativity spacetime is a four-dimensional smooth manifold with a Lorentzian metric, and the coordinate independence of physical laws is the statement that they are formulated with tensors on $M$. The diffeomorphism group of $M$ then acts as the symmetry group relating physically equivalent descriptions.

Examples

Another trivial example for a manifold is of course $R^n$.

Leads to: Curvature, Differential Forms, Differential Geometry, Fiber Bundles, Levi-Civita Connection, Morse Theory, Parallel Transport.