Manifolds

A manifold is a space that looks locally like $\mathbb{R}^n$ but may be wildly different globally. The 2-sphere is a manifold — flat to an ant walking on it, curved when viewed from space. So is the torus, the cylinder, and the surface of every smooth solid you can imagine. Manifolds are the natural domain for differential forms, integration, and the generalised Stokes' Theorem. The definition is technical, but the intuition is direct: a manifold is something you could cover with overlapping flat coordinate charts (an atlas), where adjacent charts agree smoothly on overlaps.

Definition

A topological space $M$ is an $n$-dimensional smooth manifold if it is locally Euclidean of dimension $n$ and equipped with an atlas: a collection of charts ${(U_\alpha, \phi_\alpha)}$ where each $U_\alpha \subset M$ is open, each $\phi_\alpha: U_\alpha \to \mathbb{R}^n$ is a homeomorphism onto an open subset, and where two charts overlap, the transition map $\phi_\beta \circ \phi_\alpha^{-1}$ is smooth ($C^\infty$). The charts must cover $M$.

Concrete examples

• $\mathbb{R}^n$: trivially a manifold — one chart, the identity, covers everything.
• The circle $S^1$: 1-manifold. Two charts suffice: parametrise by angle $\theta\in(-\pi,\pi)$ and $\theta\in(0,2\pi)$.
• The 2-sphere $S^2$: 2-manifold. Stereographic projection from the north pole and from the south pole gives two charts that cover $S^2$.
• The torus $T^2 = S^1\times S^1$: 2-manifold. Use a product of two circle atlases.
• Real projective space $\mathbb{RP}^n$: $n$-manifold (lines through the origin in $\mathbb{R}^{n+1}$).

Non-examples

The figure-eight curve is NOT a 1-manifold: at the crossing point, no neighbourhood looks like an interval (locally it looks like four half-lines meeting). A cone tip is NOT a smooth point of a manifold (no smooth chart). The boundary of a half-disk is a 1-manifold, but the half-disk including the boundary is a manifold with boundary, a slightly more general object.

Orientability

A manifold is orientable if the transition maps all have positive Jacobian determinant (a consistent “handedness”). Orientability is what makes integration of $n$-forms over $n$-manifolds well-defined. The most famous non-orientable surface is the Möbius strip: a normal vector slid once around the strip comes back pointing the other way. The Klein bottle and $\mathbb{RP}^2$ are non-orientable closed surfaces.

Why manifolds are the natural setting for forms

Differential forms transform predictably under smooth coordinate changes (via pullback). On a manifold, a $k$-form is defined locally in each chart, and the agreement on overlaps is automatic because of how forms pull back. This lets us integrate an $n$-form on an $n$-manifold by pulling back to $\mathbb{R}^n$ via each chart and summing with a partition of unity. The result is independent of the chart choice.

(Manifolds are inherently global geometric objects; a 2D widget cannot do them justice. The references include excellent visualisations of $S^2$, the torus, and the Möbius strip.)

Pause and think: The torus $T^2$ is a closed orientable 2-manifold. Imagine integrating a 2-form over it. What constraint does Stokes' Theorem impose on the integral of an exact form $d\alpha$? (Hint: the torus has no boundary.)

Try it

• Sketch two stereographic-projection charts covering the sphere $S^2$. What is the transition map on the overlap?
• Explain why the figure-eight (two intersecting circles) is NOT a 1-manifold but two disjoint circles ARE.
• True or false: every closed orientable 2-manifold is homeomorphic to a sphere with $g$ handles attached, for some $g \ge 0$ (the genus). Look up the classification theorem for surfaces.
• Identify the dimension and the boundary (if any) of: a closed disc, an open disc, an open Möbius strip, a torus, a circle.

A trap to watch for

Charts and their parametrisations: a chart is a map $\phi: U\to\mathbb{R}^n$, NOT a map $\mathbb{R}^n\to U$. A parametrisation is $\phi^{-1}$. Confusing the two reverses the direction of arrows everywhere — pullback becomes pushforward, etc. Garrity uses both, so read the direction of each map carefully.

What you now know

You can recognise manifolds and their dimensions, identify atlases of standard examples, and distinguish orientable from non-orientable surfaces. You know that manifolds are the natural setting for differential forms and integration. Section 6.5 puts everything together: the generalised Stokes' Theorem on manifolds.

References

• Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 6.
• Lee, J. M. (2012). Introduction to Smooth Manifolds (2nd ed.). Springer, ch. 1-4.
• Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin.
• Bachman, D. (2012). A Geometric Approach to Differential Forms (2nd ed.). Birkhäuser.
• Hubbard, J. H., Hubbard, B. B. (2015). Vector Calculus, Linear Algebra, and Differential Forms (5th ed.). Matrix Editions.