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From Charts to Smooth Maps: What a Smooth Manifold Really Is

You already met manifolds in Volume I as spaces that look locally like R^n. Now we pin down the extra data that lets you do calculus on them — a smooth atlas — and learn exactly when a map between manifolds deserves to be called smooth.

The thing you already half-know

In Volume I a manifold was a topological space M that, near every point p in M, looks like an open piece of R^n. That picture is correct, but it is missing one ingredient, and finding that ingredient is the whole job of this guide. The issue is subtle: "looks like R^n locally" only gives you a homeomorphism from a neighborhood of p to an open set in R^n — a continuous bijection with a continuous inverse. Continuity is enough to talk about open sets and limits, but it is nowhere near enough to differentiate.

Here is the trap. A homeomorphism can be brutally non-differentiable — think of the graph of the absolute-value function, which is homeomorphic to a straight line but has a corner. So "M is a topological manifold" lets you ask whether a function is continuous, but the question "is this function differentiable at p?" has no meaning yet. There is no canonical notion of a derivative on a bare topological manifold, because the homeomorphisms to R^n that we used can crinkle space arbitrarily. We need to choose better homeomorphisms — ones that agree with each other smoothly where they overlap.

Charts, atlases, and the compatibility condition

A chart on M is a pair (U, phi), where U is an open subset of M and phi: U -> R^n is a homeomorphism onto an open subset of R^n. Think of phi as assigning local coordinates: a point q in U becomes the n-tuple phi(q) = (x^1, ..., x^n). A single chart rarely covers all of M — you cannot flatten the whole sphere S^2 onto a plane without tearing it — so we collect a whole family of charts whose domains cover M. That family is an atlas, exactly like a paper atlas of the Earth: many overlapping flat pages, each distorting a different region.

The decisive idea is what happens where two charts overlap. Suppose (U, phi) and (V, psi) are two charts with U intersect V nonempty. On that overlap a point has two sets of coordinates, and the transition map psi composed with phi-inverse sends one set to the other. Crucially, this transition map goes from an open set of R^n to an open set of R^n — it is an honest map between Euclidean spaces, where we already know what "smooth" (infinitely differentiable, C-infinity) means from multivariable calculus. We call the two charts smoothly compatible if every such transition map is smooth with a smooth inverse.

Notice where the work happens: the transition map psi o phi-inverse is a map from phi(U intersect V), an open set in R^n, to psi(U intersect V), another open set in R^n. Everything stays inside Euclidean space, so "smooth" carries its ordinary calculus meaning of C-infinity. Smooth compatibility asks that both psi o phi-inverse and its reverse phi o psi-inverse be C-infinity. That single requirement, imposed on every overlapping pair, is the entire content of "giving M a calculus."

From an atlas to a smooth structure

A smooth atlas is an atlas in which every pair of charts is smoothly compatible. This is the missing ingredient: a topological manifold plus a choice of smooth atlas is a smooth manifold. Once you have it, you can finally define what a smooth function f: M -> R is — namely, f composed with phi-inverse is smooth as an ordinary function on an open set of R^n, for every chart. Compatibility is exactly the guarantee that this definition does not depend on which chart you picked, because two charts differ by a smooth transition map, and the composition of smooth maps is smooth.

One refinement matters in practice. Many different smooth atlases can describe "the same" calculus — for instance, adding one extra compatible chart changes the atlas but should not change anything geometric. So we declare two smooth atlases equivalent if their union is again a smooth atlas, and we call an equivalence class a smooth structure. Concretely you almost always work with a single convenient atlas; the equivalence class is just the bookkeeping that says your choices were inessential.

A worked example: the sphere S^2

Let us build a smooth atlas on the unit sphere S^2 in R^3 and watch compatibility do its work. The cleanest atlas uses two charts by stereographic projection. Put the source at the north pole N = (0,0,1) and project every other point onto the equatorial plane; this is a homeomorphism phi_N from S^2 minus N onto R^2. Project from the south pole S = (0,0,-1) to get a second chart phi_S on S^2 minus S. Together these two domains cover all of S^2, since the only point missing from the first is N, and it is included in the second.

Now the moment of truth: the overlap is S^2 minus the two poles, and the transition map turns out to be the inversion (u, v) -> (u, v) divided by (u^2 + v^2). That map is smooth on R^2 minus the origin — denominators never vanish there — with a smooth inverse (it is its own inverse). So the two charts are smoothly compatible, the atlas is a smooth atlas, and S^2 is a smooth manifold. The same point computed in either chart agrees up to a smooth change of coordinates, which is precisely the licence to do calculus on the sphere.

phi_N : S^2 \ {N} -> R^2      phi_S : S^2 \ {S} -> R^2     (stereographic)

overlap = S^2 \ {N, S}

transition:  phi_S o phi_N^(-1) (u, v) = (u, v) / (u^2 + v^2)
             smooth on R^2 \ {0},  and equal to its own inverse
Two stereographic charts, and a transition map that is smooth away from the origin.

Smooth maps and diffeomorphisms

With smooth structures fixed on M and N, we can finally say when a map f: M -> N is smooth. The recipe is the only one available: read f in coordinates. Pick a chart (U, phi) around p in M and a chart (V, psi) around f(p) in N; then the composite psi o f o phi-inverse is a map between open sets of Euclidean spaces, and f is smooth at p if that composite is smooth in the ordinary sense. Compatibility again saves us — change either chart and the coordinate representation changes only by smooth transition maps, so smoothness of f is a property of f alone, not of the charts.

  1. Take p in M and the target f(p) in N.
  2. Choose a chart (U, phi) around p and a chart (V, psi) around f(p), shrinking U so that f(U) lies inside V.
  3. Form the coordinate representation F = psi o f o phi-inverse, a map between open subsets of R^m and R^n.
  4. Check whether F is C-infinity by the usual multivariable rules; if it is at every point, f is smooth.

The most important smooth maps are the invertible ones. A diffeomorphism is a smooth map f: M -> N that has a smooth inverse; when one exists, M and N are indistinguishable as smooth manifolds, just as a homeomorphism makes them indistinguishable as topological spaces. Diffeomorphism is the right notion of "sameness" for everything in this rung — when we later ask whether two surfaces, two Lie groups, or two exotic spheres are "the same," we almost always mean: is there a diffeomorphism between them?

Why this matters going forward

Everything in the rest of this rung is built directly on the smooth atlas you just understood. The next guide attaches a tangent space T_p M to each point and assembles them into the tangent bundle TM; its very definition will use charts to say which curves through p "point the same way," and compatibility is what makes that well defined. After that come immersions and embeddings, the regular value theorem, Sard's theorem, and the smooth-structure-dependent invariants like the degree of a map — none of which even parse on a bare topological manifold.