Smooth Manifolds

From Continuity to Smoothness

The theory developed in the previous two pages was, by design, entirely topological. A topological manifold is built from continuity alone: a Hausdorff, second-countable space that is locally homeomorphic to \(\mathbb{R}^n\). Continuity is enough to ask when two manifolds are the same up to deformation, to count connected components, and to compare global features such as compactness and simple connectivity. It is not enough, however, to do calculus. In the entire theory of topological manifolds there is no mention of derivatives, and there cannot be one. The reason is fundamental, and we examine it first.

Why Continuity is Not Enough

The obstruction is that derivatives are not invariant under homeomorphisms. A locally Euclidean space carries, at each point, a homeomorphism between an open neighborhood and an open subset of \(\mathbb{R}^n\). This homeomorphism is the coordinate chart through which Euclidean calculus would have to be transported. But homeomorphisms are far too flexible to preserve differentiability, and a single concrete example makes this transparent.

Consider the map \[ \psi : \mathbb{R}^2 \to \mathbb{R}^2, \quad \psi(x, y) = (x^3, y^3). \] Each component is a continuous bijection of \(\mathbb{R}\) onto itself with continuous inverse \(t \mapsto t^{1/3}\), so \(\psi\) is a homeomorphism. In our chart convention, \(\psi : \mathbb{R}^2 \to \mathbb{R}^2\) qualifies as a chart on the manifold \(\mathbb{R}^2\), sending each point to its "coordinate" image. Now take the smooth function \(f : \mathbb{R}^2 \to \mathbb{R}\) defined by \(f(x, y) = x\). This function is a coordinate projection, as smooth as any function on \(\mathbb{R}^2\) could be. To read \(f\) through the chart \(\psi\), one composes with \(\psi^{-1}(u, v) = (u^{1/3}, v^{1/3})\) and computes \[ f \circ \psi^{-1} : \mathbb{R}^2 \to \mathbb{R}, \quad (u, v) \mapsto u^{1/3}. \]

This composition is not differentiable at the origin. The difference quotient that defines the partial derivative with respect to \(u\) at the origin is \(h^{1/3}/h = h^{-2/3}\), which diverges as \(h \to 0\), so that limit does not exist. The same function \(f\) that is smooth when read through the identity chart on \(\mathbb{R}^2\) fails to be even once-differentiable when read through the chart \(\psi\), even though \(\psi\) is a perfectly valid homeomorphism.

The example repays a careful reading. The function \(f\) has not changed, and neither has the underlying space \(\mathbb{R}^2\). What has changed is only the chart we use to assign coordinates. Differentiability is sensitive to the choice of chart in a way that continuity is not, and so is any notion of a derivative on the manifold. A topological manifold, in which every homeomorphism onto an open subset of \(\mathbb{R}^n\) qualifies as a chart, has no means to decide which charts give the "right" answer for differentiation.

The Strategy: An Additional Layer of Structure

The remedy is not to abandon topological manifolds but to add a layer of structure on top of them. The topological data remains unchanged. To it we adjoin a selection rule that distinguishes a privileged collection of charts on which differentiation is well-defined and chart-independent. The construction is the subject of the next section.

The extra layer is called a smooth structure, and a topological manifold endowed with one is called a smooth manifold. Before we can define it formally, we review the small piece of Euclidean calculus on which the construction rests.

Smoothness in Euclidean Space

A map \(F : U \to V\) between open subsets of Euclidean spaces is smooth (or \(C^\infty\)) if all its component functions have continuous partial derivatives of all orders. It is a diffeomorphism if it is a smooth bijection whose inverse is also smooth. Throughout this page and the rest of the manifold series, we use smooth as synonymous with \(C^\infty\). This choice avoids the bookkeeping of finitely many derivatives, and it is preserved under all the operations of the subject (compositions, restrictions, partitions of unity, and so on).

Every diffeomorphism is a homeomorphism, but the converse is false. The standard cautionary example explains why a smooth structure is a genuine extra layer rather than a consequence of the topology. The map \(\psi(x, y) = (x^3, y^3)\) is a homeomorphism of \(\mathbb{R}^2\) onto itself, and \(\psi\) itself is smooth (its components are polynomials). But the inverse \(\psi^{-1}(u, v) = (u^{1/3}, v^{1/3})\) fails to be smooth at the origin, since the partial derivatives blow up there. So \(\psi\) is a smooth homeomorphism that is not a diffeomorphism, and this is precisely the asymmetry that the smooth structure on a manifold is designed to discipline. We turn to its construction now.

Smooth Atlases and Smooth Structures

We now build the smooth structure formally. The construction has three layers: a compatibility relation between two charts, an atlas in which every pair of charts satisfies the relation, and finally a canonical maximal version of such an atlas, the smooth structure itself.

A First Attempt and Its Obstruction

The natural first attempt to define smoothness on a manifold is the following. Given a real-valued function \(f : M \to \mathbb{R}\), one would like to say that \(f\) is smooth if and only if, for every chart \((U, \varphi)\), the composite \[ f \circ \varphi^{-1} : \widehat{U} \to \mathbb{R} \] is smooth in the sense of ordinary calculus on the open set \(\widehat{U} \subseteq \mathbb{R}^n\). This definition would make smoothness on \(M\) a question about ordinary calculus on \(\mathbb{R}^n\), which is exactly what we want.

The obstruction is that the definition will only make sense if the answer does not depend on the chart used to evaluate it. The counter-example of the previous section shows that, among arbitrary topological charts, this independence fails. The chart appearing there turns a smooth function into a non-smooth one. So we are not yet entitled to declare \(f\) smooth by checking it through any single chart. Independence of the chart is the property the entire construction must secure. The route to it is to restrict from the start to a collection of charts that agree with one another on differentiability, in the following precise sense.

Transition Maps and Smooth Compatibility

Let \(M\) be a topological \(n\)-manifold, and let \((U, \varphi)\) and \((V, \psi)\) be two charts on \(M\) whose domains overlap, that is, \(U \cap V \neq \emptyset\). On the overlap \(U \cap V\), two competing systems of coordinates are available. The chart \(\varphi\) reads a point as an element of \(\varphi(U \cap V) \subseteq \mathbb{R}^n\), and \(\psi\) reads the same point as an element of \(\psi(U \cap V) \subseteq \mathbb{R}^n\). The map relating the two systems is the central object.

A transition map is, by construction, a map between open subsets of \(\mathbb{R}^n\). The set \(\varphi(U \cap V)\) is open in \(\mathbb{R}^n\) because \(\varphi\) is a homeomorphism and \(U \cap V\) is open in \(M\), and similarly for \(\psi(U \cap V)\). As a composition of two homeomorphisms, \(\psi \circ \varphi^{-1}\) is itself a homeomorphism. It is also a map between Euclidean open sets, and is therefore the kind of map to which the ordinary calculus definition of smoothness applies. Asking whether it is smooth, or even a diffeomorphism, is a question entirely within multivariable calculus.

Two charts are compatible for the purposes of differentiation precisely when their transition map is as well-behaved as Euclidean calculus allows.

Note that for a single pair of charts, smoothness of one transition map does not by itself guarantee smoothness of its inverse. The cubing map of the opening section is a concrete reminder that a smooth bijection between Euclidean open sets can have a non-smooth inverse. The diffeomorphism requirement in the definition is therefore genuinely a two-direction condition for an isolated pair of charts.

At the level of an atlas, however, the two directions need not be handled chart-pair by chart-pair. Suppose one verifies that for every ordered pair \(((U, \varphi), (V, \psi))\) of charts in a candidate atlas the forward transition \(\psi \circ \varphi^{-1}\) is smooth. Then for any chart pair the reverse transition \(\varphi \circ \psi^{-1}\) is also smooth. No geometric or analytic theorem is needed for this, because \(\varphi \circ \psi^{-1}\) is itself the forward transition of the ordered pair \(((V, \psi), (U, \varphi))\), already covered by the enumeration.

The total amount of smoothness verification is the same as checking both directions for each unordered pair. What the atlas structure does is reorganize the work, so that one never has to address diffeomorphism for a chart pair as a separate task beyond verifying forward smoothness of every ordered pair.

Smooth Atlases

Recall that an atlas for a topological manifold is a collection of charts whose domains cover the manifold. A smooth atlas is the natural strengthening in which every pair of charts passes the compatibility test.

Two practical remarks. First, the compatibility condition needs to be checked only on pairs of charts whose domains actually overlap. Pairs with disjoint domains are smoothly compatible by definition. Second, by the observation above, the verification of smooth compatibility for the entire atlas reduces to checking forward smoothness of the transition map for each ordered pair of overlapping charts. The diffeomorphism condition for each chart pair then holds because both ordered directions appear in this enumeration.

We have not yet exhibited a single example of a smooth atlas. The topological atlases of \(\mathbb{R}^n\), spheres, projective spaces, and tori from the previous pages each give rise to smooth atlases, but the verification of smooth compatibility is a separate matter that we postpone until the standard smooth structures section. For now we treat the abstract notion of smooth atlas as the data we may assume given.

Many Atlases, One Structure

A single manifold admits many different smooth atlases. On \(\mathbb{R}^n\), for instance, the one-chart atlas \(\{(\mathbb{R}^n, \mathrm{id})\}\) is a smooth atlas, but so is the collection of all charts \((U, \mathrm{id}|_U)\) for arbitrary open \(U \subseteq \mathbb{R}^n\), and so is the collection obtained from either of these by applying diffeomorphisms to the codomain. These atlases are not literally equal as sets of charts, but they all declare the same functions \(\mathbb{R}^n \to \mathbb{R}\) to be smooth. A definition of "smooth manifold" that fixed one particular atlas would be artificial. The geometry that matters is invariant under enlarging the atlas, as long as the enlargement remains smoothly compatible everywhere.

The canonical way to remove this ambiguity is to enlarge any given smooth atlas until it cannot be enlarged any further. This produces a distinguished representative, the maximal smooth atlas, which we will take as the definitional object.

An equivalent formulation, which is often more conceptually appealing, proceeds by an equivalence relation. We declare two smooth atlases equivalent when their union is again a smooth atlas, and we define a smooth structure to be an equivalence class. The maximal-atlas formulation and the equivalence-class formulation produce the same objects in bijection, since the equivalence class containing \(\mathcal{A}\) corresponds to the unique maximal atlas containing \(\mathcal{A}\), as we will prove below. We adopt the maximal-atlas formulation as primary, because it makes the smooth structure a concrete set of charts rather than an abstract quotient.

Two cautions of substance. First, the smooth structure is genuinely additional data. A given topological manifold may carry several distinct smooth structures, and the small-dimensional case will be witnessed concretely in the standard smooth structures section, when we exhibit two inequivalent smooth structures on the underlying topological space \(\mathbb{R}\).

Second, the existence of any smooth structure at all is not automatic. There are topological manifolds that admit no smooth structure whatsoever. The first such example was a compact \(10\)-dimensional manifold discovered by Michel Kervaire in 1960. Its construction is far beyond the scope of the present development, but its existence shows that the smoothability of a topological manifold is a genuine, nontrivial property rather than a foregone conclusion.

From a Smooth Atlas to a Smooth Structure

Defining a smooth structure as a maximal smooth atlas has a practical disadvantage. A maximal atlas is enormous, and it contains every chart on every open subset that is smoothly compatible with every other admitted chart. We never construct a maximal atlas by listing its charts. The following proposition supplies the working tool. Any smooth atlas determines a unique smooth structure containing it, so to specify a smooth structure it suffices to specify any smooth atlas, however small.

With this proposition in hand, the working strategy is clear. To define a smooth structure on a topological manifold, one specifies any smooth atlas, and the smooth structure is taken to be the (unique) maximal smooth atlas it determines. The remainder of this page will follow exactly this pattern. We will define each forthcoming smooth structure by writing down a small, explicit smooth atlas, and the smooth structure will be understood to be its maximal extension.

Local Coordinate Representations

Smooth Charts and Smooth Coordinate Maps

Let \(M\) be a smooth manifold with smooth structure \(\mathcal{A}\). The charts of \(M\) on which differentiation can be done without ambiguity are precisely the elements of \(\mathcal{A}\).

The shape-based refinements of a chart introduced in the topological setting transfer immediately to the smooth setting.

The terminology mirrors the topological case, with the single added requirement that the chart in question is an element of the smooth structure.

Regular Coordinate Balls

Many arguments in differential geometry require a coordinate ball together with a slightly larger smooth chart in which it sits with compact closure. Packaging this pair into a single named object proves convenient.

The definition packages three pieces of geometric information. The set \(B\) itself is a smooth coordinate ball of radius \(r\). Its closure \(\overline{B}\) corresponds under \(\varphi\) to the closed ball of the same radius, and \(\overline{B}\) is contained in a strictly larger smooth coordinate ball \(B'\) of radius \(r'\).

The closure here is unambiguous. Because \(\overline{B_r(0)}\) is compact in \(\mathbb{R}^n\), its homeomorphic image is compact in the smooth coordinate ball \(B'\), and since \(M\) is Hausdorff this image is closed in \(M\). The closure of \(B\) in \(M\) therefore coincides with its closure in \(B'\). Consequently \(\overline{B}\) is compact in \(M\), and every regular coordinate ball is therefore precompact in \(M\) in the sense introduced in the previous page on topological properties of manifolds.

The benefit of having a smooth coordinate ball sit inside a larger smooth chart with compact closure is technical but pervasive. Cutoff functions can be constructed on \(\overline{B}\) using the room provided by \(B'\), compactness allows uniform estimates, and the larger chart provides a smooth context in which to perform local arguments without boundary effects. The next proposition states that every smooth manifold admits a countable basis of such regular coordinate balls. This proposition is the smooth refinement of the basis-of-precompact-coordinate-balls result for topological manifolds.

The proof sketch is faithful to the structure of the topological version. The only smooth content is verifying that the charts produced by translation remain inside the smooth structure, which is automatic because translations are diffeomorphisms of \(\mathbb{R}^n\) onto itself.

Reading Through a Chart: The Identification Habit

The framework so far is precise but typographically cumbersome. Every statement about a point \(p \in M\) seems to require explicit reference to a chart \(\varphi\) and to the image \(\varphi(p) \in \mathbb{R}^n\). In practice, mathematicians working with smooth manifolds adopt a much lighter notational habit, and it is essential to understand and internalize it before proceeding.

Once a smooth chart \((U, \varphi)\) is fixed, the coordinate map \(\varphi : U \to \widehat{U} \subseteq \mathbb{R}^n\) is a homeomorphism, and we can regard it as a temporary identification between the open subset \(U \subseteq M\) and the open subset \(\widehat{U} \subseteq \mathbb{R}^n\). While we are working inside this chart, we will simultaneously think of \(U\) as a piece of the abstract manifold and as a piece of Euclidean space.

A point \(p \in U\) is represented by its coordinates \[ (x^1, \ldots, x^n) = \varphi(p), \] and the standard practice is to say that "\((x^1, \ldots, x^n)\) is the (local) coordinate representation of \(p\)," or, more briefly, that "\(p = (x^1, \ldots, x^n)\) in local coordinates." Within the chart, the coordinate map \(\varphi\) is suppressed from the notation. Effectively, one pretends that \(\varphi\) is the identity map.

This habit takes practice to read fluently but yields enormous notational economy. Two cautions accompany it. First, the identification is purely local. It depends on the choice of chart and is valid only on \(U\). A point in the overlap of two charts has two coordinate representations, one for each chart, related by the transition map. Second, the identification suppresses information about the global structure of \(M\), since a chart never sees the manifold beyond its own domain. The geometric content that distinguishes a manifold from Euclidean space lives precisely in how distinct charts must be glued together to recover the whole.

Local Coordinate Representations of Functions

The identification through a chart extends naturally to functions on the manifold. Given a function \(f : M \to \mathbb{R}\) and a smooth chart \((U, \varphi)\), the composition \[ \widehat{f} := f \circ \varphi^{-1} : \widehat{U} \to \mathbb{R} \] is a real-valued function on the open subset \(\widehat{U} \subseteq \mathbb{R}^n\). This is called the local coordinate representation of \(f\) with respect to the chart \((U, \varphi)\), and it expresses \(f\) as an ordinary function on Euclidean space in terms of the local coordinates \((x^1, \ldots, x^n)\).

The earlier motivating question asked when \(f\) is smooth on \(M\). That question is answered later in the manifold series in terms of \(\widehat{f}\). The formal definition will declare \(f\) smooth at \(p\) when its local representation \(\widehat{f}\) is smooth in the ordinary Euclidean sense near \(\varphi(p)\), and the smooth compatibility built into the structure will be exactly what ensures that this property is independent of the chart used to test it. The present subsection only sets up the notation \(\widehat{f}\) on which that future definition will rest.

Maps between manifolds admit local coordinate representations of the same kind. If \(F : M \to N\) is a map of smooth manifolds and we choose smooth charts \((U, \varphi)\) on \(M\) and \((V, \psi)\) on \(N\) with \(F(U) \subseteq V\), then the composition \[ \widehat{F} := \psi \circ F \circ \varphi^{-1} : \widehat{U} \to \widehat{V} \] is a map between Euclidean open sets, and is the local coordinate representation of \(F\).

All questions about \(F\) that are local in nature are reduced through \(\widehat{F}\) to ordinary calculus, and the choice of charts ceases to matter as soon as one shows the relevant property is invariant under change of chart. Such questions include smoothness, differentials, and invertibility. The formal definition of smoothness for maps between manifolds is taken up later in the manifold series. Here we only record the notational machinery on which it rests.

An Example from Multivariable Calculus: Polar Coordinates

The identification habit is not new. Every reader has practiced it, under a different name, when using polar coordinates in the plane.

Let \(U = \{(x, y) \in \mathbb{R}^2 : x \gt 0\}\), the open right half-plane in \(\mathbb{R}^2\) regarded as a smooth manifold with its standard smooth structure (to be defined formally in the standard smooth structures section). The relation \[ (x, y) = (r \cos\theta, r \sin\theta) \] defines polar coordinates \((r, \theta)\) on \(U\), with \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan(y/x)\) for points in \(U\).

The use of \(\arctan(y/x)\) requires a brief justification. In general the formula \(\theta = \arctan(y/x)\) determines \(\theta\) only up to addition of integer multiples of \(\pi\), because \(\tan(\theta) = \tan(\theta + \pi)\). The function \(\arctan(y/x)\) is well-defined as a single-valued function only on a half-plane that does not cross the \(y\)-axis. On the right half-plane \(U = \{x \gt 0\}\), the angle \(\theta\) lies in the open interval \((-\pi/2, \pi/2)\). Moreover, the quotient \(y/x\) is a smooth function on \(U\) (a rational function with non-vanishing denominator), and \(\arctan : \mathbb{R} \to (-\pi/2, \pi/2)\) is itself smooth, so the composition \(\theta(x, y) = \arctan(y/x)\) is a smooth function on \(U\). The formula \(\theta = \arctan(y/x)\) thus produces a single, smooth, well-defined angle. Restricting to \(U\) is exactly the device that makes the polar chart globally well-defined.

The map \[ \varphi : U \to (0, \infty) \times \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right), \quad (x, y) \mapsto (r, \theta), \] is a smooth coordinate map with respect to the standard smooth structure on \(\mathbb{R}^2\). Both \(r\) and \(\theta\) are smooth functions of \((x, y)\) on \(U\), and the inverse map \((r, \theta) \mapsto (r\cos\theta, r\sin\theta)\) is likewise smooth. The pair \((U, \varphi)\) is thus an example of a smooth chart, and the standard practice of writing "a point of \(U\) has polar coordinates \((r, \theta)\)" instead of writing "the point \(p \in U\) satisfies \(\varphi(p) = (r, \theta)\)" is precisely the identification habit at work. Other polar coordinate charts can be obtained by applying the same recipe on other open half-planes. The formula \(\theta = \arctan(y/x)\) cannot simply be carried over to an arbitrary open subset of \(\mathbb{R}^2 \setminus \{0\}\), since it is undefined where \(x = 0\) and elsewhere recovers the angle only up to a multiple of \(\pi\), so a branch of the angle must be chosen afresh in each case.

The example also shows that a single smooth manifold typically admits many smooth charts on a single open set, related by smooth transition maps. On \(U\), the identity chart \((U, \mathrm{id})\) and the polar chart \((U, \varphi)\) are smoothly compatible. The transition \(\varphi \circ \mathrm{id}^{-1} = \varphi\) is smooth, and so is its inverse \((r, \theta) \mapsto (r\cos\theta, r\sin\theta)\). The flexibility to switch between such coordinate representations, according to whichever makes a given calculation easiest, is one of the central practical techniques of manifold theory.

The Einstein Summation Convention

Before turning to concrete examples of smooth manifolds, we pause to introduce a notational convention that pervades the rest of the manifold series. The convention is purely typographical, but it is used so universally in differential geometry and applied differential geometry that fluency with it is a prerequisite for reading the literature. Robotics, general relativity, and geometric deep learning all rely on it. Introducing it here, before the substantive work begins, saves having to translate notation later.

The Problem: Proliferation of Summations

The identification habit of the previous section already foreshadowed the issue. A point on a smooth \(n\)-manifold, read through a chart, carries \(n\) coordinates \(x^1, \ldots, x^n\). A vector at that point (when we eventually define such a thing) will be expanded in terms of \(n\) basis vectors. A change of coordinates between two overlapping charts is, in general, a smooth but nonlinear map \[ \widetilde{x}^j = \widetilde{x}^j(x^1, \ldots, x^n) \] on the Euclidean side. Only its differential at each point, the Jacobian matrix, is linear. Each of these objects involves a sum from \(1\) to \(n\) over some index, and longer expressions accumulate multiple such sums.

The simplest case where the convention becomes essential is the linear one. A finite-dimensional vector space \(V\) is itself a smooth manifold, as a later section of this page will show. In such a space, expressing a vector in a basis or relating coordinates in two different bases produces formulas of the form \[ v = \sum_{i=1}^{n} v^i E_i, \quad \widetilde{x}^j = \sum_{i=1}^{n} A^j_i x^i, \] where \((A^j_i)\) is the matrix of an invertible linear change of coordinates. Such linear formulas appear frequently. They occur in the vector-space examples, in tangent spaces (a linear structure attached to each point of a manifold), and in the chain rule for nonlinear coordinate changes. Stringing several of them together becomes typographically unmanageable. Einstein's solution, introduced in his work on general relativity, was to drop the summation sign and read the implied sum off from the index pattern.

The Convention

Under this convention, the expansion of a vector \(v\) in a basis is written \[ v = v^i E_i, \] with the right-hand side understood to mean \(\sum_{i=1}^{n} v^i E_i\). The linear change-of-basis formula above becomes \[ \widetilde{x}^j = A^j_i x^i, \] where \(i\) is summed from \(1\) to \(n\). Composition of two such linear transformations is similarly compact: \[ \widetilde{\widetilde{x}}^k = B^k_j \widetilde{x}^j = B^k_j A^j_i x^i, \] where both \(i\) and \(j\) are summed.

Index Position

The convention works only because index position carries information. Two conventions must be respected consistently:

• Basis vectors.
  We always write these with lower indices, for example \(E_1, \ldots, E_n\).
• Components of a vector with respect to a basis.
  We always write these with upper indices, for example \(v^1, \ldots, v^n\).

Coordinates of a point \((x^1, \ldots, x^n) \in \mathbb{R}^n\) are written with upper indices. On \(\mathbb{R}^n\) the reason is the consistency rule above. A point of \(\mathbb{R}^n\) is identifiable with its position vector relative to the origin. Its coordinates therefore coincide with the components of that vector with respect to the standard basis and sit in the same upper-index position.

On a general smooth manifold a point is not a vector. There is no canonical origin and no canonical position vector. The upper-index convention for local coordinates \((x^1, \ldots, x^n) = \varphi(p)\) is therefore adopted by extension, but its deeper justification will come later in the manifold series, when tangent vectors are introduced. Under a change of coordinates on a manifold the coordinates themselves transform nonlinearly: \(\widetilde{x}^j = \widetilde{x}^j(x^1, \ldots, x^n)\), a smooth function of the \(x^i\) that need not be linear. What transforms linearly is the differential \(d\widetilde{x}^j = \tfrac{\partial \widetilde{x}^j} {\partial x^i} dx^i\), and likewise the components of a tangent vector \(v^i\) transform as \(\widetilde{v}^j = \tfrac{\partial \widetilde{x}^j}{\partial x^i} v^i\). The upper-index convention for coordinates is what makes the indices on \(dx^i\) and on tangent vector components line up with these transformation laws. The upper position is a fact about notation, not a claim that coordinates themselves are vector components. For now, the convention is taken as given. The notational asymmetry between coordinates \(x^i\) and basis vectors \(E_j\) is what makes the implicit summation in \(v = v^i E_i\) unambiguous and grammatically well-formed.

Dummy Indices

An index that is summed over has no independent meaning. Its name can be replaced by any other unused name without changing the value of the expression. Such an index is called a dummy index. For example, \[ x^i E_i = x^j E_j = x^k E_k, \] all denoting the same sum. The names \(i\), \(j\), \(k\) are placeholders. What matters is the pattern of paired upper-lower occurrences.

By contrast, an index that appears only once in an expression is called a free index. A free index carries genuine meaning, and its name must not be reused in the same expression to refer to a different quantity. In the formula \(\widetilde{x}^j = A^j_i x^i\), the index \(i\) is dummy (summed) and the index \(j\) is free (refers to a specific component on each side). Writing \(\widetilde{x}^i = A^i_i x^i\) would be a notational error. The name \(i\) would appear as both a free index and a dummy index, which the convention cannot disambiguate.

Scope of the Convention

From here onward we adopt the Einstein convention by default throughout the manifold series. An explicit \(\sum\) sign will appear only in two situations: when the indices do not follow the upper-lower pairing rule (so the convention does not apply), and when additional clarity is desired in an example that is being introduced for the first time.

All this may seem awkward at first. Keeping track of which indices go up and which go down is an extra layer of bookkeeping, and the disappearance of the summation sign can feel like a loss of grammar rather than a gain in economy. The discomfort passes with practice. In combination with the upper-lower convention, the implicit-summation rule makes coordinate changes, tensor contractions, and covariant derivatives far less typographically dense than they would otherwise be. All three are formal objects that we meet later in the manifold series. The index pattern itself also becomes a useful diagnostic for whether an expression makes geometric sense at all.

Standard Smooth Structures

Discrete Spaces as 0-Dimensional Smooth Manifolds

The smallest example of all is the case of dimension zero, where the manifold framework collapses to something almost trivial. A topological manifold of dimension zero is a Hausdorff, second-countable, locally Euclidean space whose local model is the single-point space \(\mathbb{R}^0 = \{0\}\). As a one-point space, \(\mathbb{R}^0\) has only two subsets at all (the empty set and \(\{0\}\) itself), and both are necessarily open. Being homeomorphic to a neighborhood of a point in \(\mathbb{R}^0\) therefore forces the neighborhood to be a singleton, and every point of the manifold is open as a singleton. The space is discrete, and second-countability then makes it countable.

On such a space there is essentially nothing to choose. At each point \(p \in M\), the only chart available is \(\varphi : \{p\} \to \{0\}\), the unique map between two one-point sets. Any two such charts have domains that are either equal (in which case their transition map is the identity on \(\mathbb{R}^0\), trivially smooth) or disjoint (in which case the smooth compatibility condition is vacuous). The collection of all such charts is therefore automatically a smooth atlas, and the smooth structure it generates is the only possible one. Every 0-dimensional topological manifold carries a unique smooth structure.

Euclidean Space as a Smooth Manifold

The Euclidean space \(\mathbb{R}^n\), the prototype of all manifolds, carries a canonical smooth structure that requires no choice.

The single-chart atlas \(\mathcal{A}_{\mathrm{std}}\) is trivially a smooth atlas. With only one chart, there are no nonempty overlaps to check, so the smooth compatibility condition holds vacuously. By the generation proposition, it determines a unique smooth structure on \(\mathbb{R}^n\). Throughout the manifold series, and throughout the literature, the convention is firm. Unless an alternative smooth structure is explicitly specified, \(\mathbb{R}^n\) carries the standard smooth structure.

With respect to this smooth structure, the smooth charts on \(\mathbb{R}^n\) can be characterized concretely. A chart \((U, \varphi)\) is smooth precisely when \(\varphi : U \to \widehat{U} \subseteq \mathbb{R}^n\) is a diffeomorphism in the ordinary sense of multivariable calculus. The reason is direct. A chart \((U, \varphi)\) belongs to the standard smooth structure if and only if it is smoothly compatible with \((\mathbb{R}^n, \mathrm{id})\). Compatibility means that the transition maps \(\mathrm{id} \circ \varphi^{-1} = \varphi^{-1}\) and \(\varphi \circ \mathrm{id}^{-1} = \varphi\) (read on the appropriate overlap, which is just \(U\) itself) are both smooth, that is, that \(\varphi\) is a diffeomorphism. So the smooth charts of \(\mathbb{R}^n\) are exactly its Euclidean diffeomorphisms onto open subsets, and the polar chart discussed earlier on this page is one such.

Another Smooth Structure on \(\mathbb{R}\)

Consider the real line \(\mathbb{R}\) regarded as a topological manifold, and define the map \[ \psi : \mathbb{R} \to \mathbb{R}, \quad \psi(x) = x^3. \] The map \(\psi\) is continuous and bijective, with continuous inverse \(\psi^{-1}(y) = y^{1/3}\), hence a homeomorphism. The single-chart atlas \(\{(\mathbb{R}, \psi)\}\) is therefore an atlas in the topological sense and, by the same vacuous argument as for \(\mathcal{A}_{\mathrm{std}}\), a smooth atlas. Call its determined smooth structure \(\mathcal{A}_\psi\).

The natural question is whether \(\mathcal{A}_\psi\) is the same as the standard smooth structure \(\mathcal{A}_{\mathrm{std}}\) on \(\mathbb{R}\). By part (2) of the generation proposition, the two structures are equal if and only if the charts \((\mathbb{R}, \mathrm{id})\) and \((\mathbb{R}, \psi)\) are smoothly compatible. They are not. The transition map from \(\psi\) to \(\mathrm{id}\) is \[ \mathrm{id} \circ \psi^{-1} : \mathbb{R} \to \mathbb{R}, \quad y \mapsto y^{1/3}, \] and this map fails to be smooth at the origin. Its difference quotient at the origin is \(h^{1/3}/h = h^{-2/3}\), which diverges as \(h \to 0\), so the limit defining the derivative there does not exist. So \((\mathbb{R}, \mathrm{id})\) and \((\mathbb{R}, \psi)\) are not smoothly compatible, and \(\mathcal{A}_\psi\) differs from \(\mathcal{A}_{\mathrm{std}}\).

The proof is the verification just given. A reader meeting this example for the first time should pause to note what is and what is not being claimed.

The cubing map appeared in the opening section of this page in the two-dimensional form \(\psi(x,y) = (x^3, y^3)\), but in a different role. There it served as a smooth homeomorphism whose non-smooth inverse \(\psi^{-1}(u,v) = (u^{1/3}, v^{1/3})\) made an otherwise smooth function look non-smooth when read through it. That example showed that smoothness is not a homeomorphism-invariant property, and it motivated the introduction of smooth structures in the first place. Here the one-dimensional version \(\psi(x) = x^3\) plays a different structural role. It defines a smooth chart of a smooth structure on \(\mathbb{R}\), but not of the standard one.

The reappearance is not a coincidence. Both phenomena trace back to the failure of the cube-root map \(t \mapsto t^{1/3}\) to be smooth at the origin. The two situations are worth comparing. In the opening section, the non-smoothness of \(\psi^{-1}\) disqualified \(\psi\) from sitting in any smooth structure compatible with the standard smooth structure on \(\mathbb{R}^2\). Here the same failure is what keeps \(\mathcal{A}_\psi\) from equaling \(\mathcal{A}_{\mathrm{std}}\) on \(\mathbb{R}\).

Three final observations are worth recording. First, although \(\mathcal{A}_\psi\) and \(\mathcal{A}_{\mathrm{std}}\) are distinct as smooth structures, this distinctness has a sharper expression than the abstract inequality of atlases. Viewed as a map between the two smooth manifolds, the set-theoretic identity \(x \mapsto x\) fails to be a diffeomorphism in either direction. Read \(\mathrm{id} : (\mathbb{R}, \mathcal{A}_\psi) \to (\mathbb{R}, \mathcal{A}_{\mathrm{std}})\) through the chart \(\psi\) on the domain and the chart \(\mathrm{id}_\mathbb{R}\) on the codomain. The local coordinate representation is \[ \mathrm{id}_\mathbb{R} \circ \mathrm{id} \circ \psi^{-1}(y) = \psi^{-1}(y) = y^{1/3}, \] the cube-root map, which is not smooth at the origin. This direction of the identity is not even a smooth map, let alone a diffeomorphism.

The reverse direction \(\mathrm{id} : (\mathbb{R}, \mathcal{A}_{\mathrm{std}}) \to (\mathbb{R}, \mathcal{A}_\psi)\), read through \(\mathrm{id}_\mathbb{R}\) on the domain and \(\psi\) on the codomain, has local representation \[ \psi \circ \mathrm{id} \circ \mathrm{id}_\mathbb{R}^{-1}(x) = \psi(x) = x^3, \] which is smooth. Its inverse is the previous direction, which is not smooth, so this direction is a smooth map without being a diffeomorphism. Either way, the set-theoretic identity fails to be a diffeomorphism between the two smooth manifolds. This asymmetric failure is the geometric content of "distinct smooth structure": same underlying set, same topology, but no diffeomorphism via the identity.

Second, although \(\mathcal{A}_\psi\) and \(\mathcal{A}_{\mathrm{std}}\) are distinct as smooth structures, the two resulting smooth manifolds turn out to be diffeomorphic, not via the identity but via a different map. The map \[ \Phi : (\mathbb{R}, \mathcal{A}_\psi) \to (\mathbb{R}, \mathcal{A}_{\mathrm{std}}), \quad \Phi(x) = x^3, \] is a diffeomorphism between the two smooth manifolds. To check this, one reads \(\Phi\) through the chart \(\psi\) on the domain and the chart \(\mathrm{id}\) on the codomain. The local coordinate representation is \[ \mathrm{id} \circ \Phi \circ \psi^{-1}(y) = \Phi(y^{1/3}) = (y^{1/3})^3 = y, \] the identity on \(\mathbb{R}\), which is smooth. The inverse map \(\Phi^{-1}(x) = x^{1/3}\) has coordinate representation \(\psi \circ \Phi^{-1} \circ \mathrm{id}^{-1}(x) = (x^{1/3})^3 = x\), again the identity, also smooth. So although \(\mathcal{A}_\psi\) is not the standard smooth structure on \(\mathbb{R}\), the smooth manifolds \((\mathbb{R}, \mathcal{A}_\psi)\) and \((\mathbb{R}, \mathcal{A}_{\mathrm{std}})\) are diffeomorphic. The distinction between "distinct smooth structure" and "non-diffeomorphic smooth manifold" is therefore real and important. The real line admits many smooth structures, but only one diffeomorphism class of smooth manifold structure, by the classification of smooth \(1\)-manifolds.

Third, the technique generalizes. By composing with non-smooth homeomorphisms of \(\mathbb{R}\), one can construct infinitely many distinct smooth structures on the same underlying topological \(\mathbb{R}\), all mutually diffeomorphic as smooth manifolds. The higher-dimensional analogues are far more delicate. In dimension four, the standard topological space \(\mathbb{R}^4\) carries uncountably many smooth structures that are not pairwise diffeomorphic. The proof of that famous result requires the full machinery of gauge theory and lies far beyond our present scope.

Vector Spaces and Matrix Spaces

Finite-Dimensional Real Vector Spaces

Let \(V\) be a finite-dimensional real vector space of dimension \(n\). Before \(V\) can be regarded as a smooth manifold, it must be regarded as a topological manifold. To this end, recall that any norm \(\|\cdot\|\) on \(V\) determines a topology, and that on a finite-dimensional space all norms are equivalent. Any two norms induce the same topology, so the topology on \(V\) is canonical, independent of the choice of norm. In this topology \(V\) must be shown to be a topological \(n\)-manifold, that is, Hausdorff, second-countable, and locally Euclidean. The verification is most efficiently obtained from a single observation: any choice of basis gives a linear bijection \(\mathbb{R}^n \to V\) which is automatically a homeomorphism. Hausdorff, second-countability, and the locally Euclidean property are all topological invariants that transfer along homeomorphisms from \(\mathbb{R}^n\). We make this construction explicit, since the same map will also produce the smooth chart.

Choose an ordered basis \((E_1, \ldots, E_n)\) of \(V\), and define the basis isomorphism \[ E : \mathbb{R}^n \to V, \quad E(x) = x^i E_i. \] Here the Einstein summation convention is in force. The right-hand side denotes \(\sum_{i=1}^{n} x^i E_i\). The map \(E\) is a linear bijection, and by general principles its inverse \(E^{-1} : V \to \mathbb{R}^n\) is also linear. Both maps are continuous with respect to the standard Euclidean topology on \(\mathbb{R}^n\) and the norm topology on \(V\), since any linear map on a finite-dimensional normed space is bounded and therefore continuous. In finite dimension the choice of norm on either side is immaterial. So \(E\) is a homeomorphism, which both transfers Hausdorffness and second-countability from \(\mathbb{R}^n\) to \(V\) and exhibits \(V\) as locally (in fact globally) Euclidean. The single-chart collection \(\{(V, E^{-1})\}\) is therefore a topological atlas on \(V\), with \(E^{-1}\) sending each vector to its coordinate \(n\)-tuple in the chosen basis.

The question is whether the smooth structure determined by this atlas depends on the basis. Suppose \((\widetilde{E}_1, \ldots, \widetilde{E}_n)\) is another ordered basis of \(V\), and let \(\widetilde{E} : \mathbb{R}^n \to V\), \(\widetilde{E}(x) = x^j \widetilde{E}_j\), be the corresponding basis isomorphism. Express the old basis in terms of the new. There exist scalars \(A^j_i\) (\(i, j = 1, \ldots, n\)) such that \[ E_i = A^j_i \widetilde{E}_j, \] and the matrix \((A^j_i)\) is invertible because both \((E_i)\) and \((\widetilde{E}_j)\) are bases. Substituting, \[ E(x) = x^i E_i = x^i A^j_i \widetilde{E}_j = (A^j_i x^i) \widetilde{E}_j, \] so the transition map between the two charts is \[ \widetilde{E}^{-1} \circ E : \mathbb{R}^n \to \mathbb{R}^n, \quad (\widetilde{E}^{-1} \circ E)(x) = \widetilde{x}, \quad \widetilde{x}^j = A^j_i x^i. \] This transition map is an invertible linear map of \(\mathbb{R}^n\), hence a diffeomorphism in the ordinary sense of multivariable calculus. The two charts \((V, E^{-1})\) and \((V, \widetilde{E}^{-1})\) are therefore smoothly compatible. By symmetry the same holds for any pair of basis-induced charts, and the collection of all such charts is a smooth atlas.

The basis-induced charts are all smoothly compatible with each other, so any one of them generates the same maximal atlas. The smooth structure does not depend on which basis was used to define it.

When \(V = \mathbb{R}^n\) and the basis is the standard basis \((e_1, \ldots, e_n)\), the basis isomorphism is the identity, and the standard smooth structure on \(V\) coincides with the standard smooth structure on \(\mathbb{R}^n\) defined earlier.

Matrix Spaces

The space of \(m \times n\) real matrices, denoted \(M(m \times n, \mathbb{R})\), is a finite-dimensional real vector space of dimension \(mn\) under entrywise matrix addition and scalar multiplication. By the construction just given, it carries a canonical standard smooth structure as a smooth \(mn\)-manifold.

In practice this smooth manifold is often identified with \(\mathbb{R}^{mn}\) by the obvious mapping. Write a matrix \(A = (a_{ij})\) and concatenate its entries into a single \(mn\)-tuple, for example by listing them row by row. This identification is an isomorphism of vector spaces, and is therefore a diffeomorphism of smooth manifolds. The choice between thinking of \(M(m \times n, \mathbb{R})\) as a matrix space or as \(\mathbb{R}^{mn}\) is a matter of notational convenience for the problem at hand.

The complex case is similar. The space \(M(m \times n, \mathbb{C})\) of \(m \times n\) complex matrices is a real vector space of dimension \(2mn\). Each complex entry contributes two real degrees of freedom, one for its real part and one for its imaginary part. The space is therefore a smooth manifold of dimension \(2mn\), again by the basis-induced construction.

For square matrices the notation is abbreviated. When \(m = n\), the spaces \(M(n \times n, \mathbb{R})\) and \(M(n \times n, \mathbb{C})\) are written as \(M(n, \mathbb{R})\) and \(M(n, \mathbb{C})\), respectively. The smooth manifold \(M(n, \mathbb{R})\) is the ambient space inside which all of the classical matrix groups are defined, and the basis-induced smooth structure on it is the one with respect to which matrix multiplication, determinant, trace, and inverse will turn out to be smooth maps. The verification of smoothness for these operations belongs to a later page in the manifold series, where smooth maps between smooth manifolds are defined and studied.

Spaces of Linear Maps

The space of linear maps between two finite-dimensional real vector spaces is also, naturally, a smooth manifold. Let \(V\) and \(W\) be finite-dimensional real vector spaces of dimensions \(n\) and \(m\) respectively, and let \(L(V; W)\) denote the set of linear maps from \(V\) to \(W\). Under pointwise addition and scalar multiplication, \(L(V; W)\) is itself a real vector space, of dimension \(mn\). By the construction given above for finite-dimensional vector spaces, it carries a canonical standard smooth structure as a smooth \(mn\)-manifold.

A concrete chart is obtained by choosing bases. Fix ordered bases \((E_1, \ldots, E_n)\) for \(V\) and \((F_1, \ldots, F_m)\) for \(W\). A linear map \(T \in L(V; W)\) is determined by its action on the basis of \(V\), and writing \[ T(E_i) = T^j_i \, F_j \] (Einstein convention, with \(j\) summed from \(1\) to \(m\)) assigns to \(T\) the matrix of scalars \((T^j_i) \in M(m \times n, \mathbb{R})\). This assignment is a linear isomorphism \(L(V; W) \cong M(m \times n, \mathbb{R})\), and is therefore a diffeomorphism of smooth manifolds. The choice of bases affects the explicit form of the isomorphism but not the smooth structure on \(L(V; W)\), which is intrinsic.

The example is more useful than its brevity might suggest. The parameter space of a single linear layer in a neural network is, after fixing input and output bases, a copy of \(L(V; W)\). The space of linear operators on the state space of a quantum system is \(L(V; V)\) for an appropriate complex vector space \(V\). When smooth maps are taken up later in the manifold series, the differential \(dF_p\) of a smooth map \(F : M \to N\) at a point \(p \in M\) will be a linear map between tangent spaces and will therefore live in such an \(L(V; W)\). The smooth structure on \(L(V; W)\) is what makes it sensible to speak of \(dF_p\) as varying smoothly with \(p\).

Open Submanifolds, \(GL(n,\mathbb{R})\), and Smooth Graphs

The final family of examples collected on this page rests on a single observation that has accompanied us since the opening of the manifold series: open subsets of manifolds are themselves manifolds, in a natural way. Lifting this construction from the topological setting to the smooth one provides immediate access to two of the most important smooth manifolds in the subject, the general linear group and the space of full-rank matrices. The same lifting gives the smooth refinement of the graph construction used in the topological case.

Open Submanifolds

Let \(M\) be a smooth \(n\)-manifold and let \(U \subseteq M\) be an open subset. The topological result already established says that \(U\), equipped with the subspace topology, is itself a topological \(n\)-manifold. We promote it to a smooth manifold by restricting the smooth charts of \(M\) to \(U\).

The verification is direct. Every point \(p \in U\) is contained in the domain of some smooth chart \((W, \varphi) \in \mathcal{A}\) of \(M\). We claim that the restriction \((W \cap U, \varphi|_{W \cap U})\) lies in \(\mathcal{A}_U\). The restricted chart is smoothly compatible with every chart \((V, \psi) \in \mathcal{A}\). The transition map \(\psi \circ (\varphi|_{W \cap U})^{-1}\) is just the original transition \(\psi \circ \varphi^{-1}\) restricted to the open subset \(\varphi(W \cap U \cap V) \subseteq \varphi(W \cap V)\), and a smooth function restricted to an open subdomain remains smooth. The same holds for the reverse direction. By maximality of \(\mathcal{A}\), the restricted chart belongs to \(\mathcal{A}\), and since its domain \(W \cap U\) lies in \(U\), it belongs to \(\mathcal{A}_U\). The charts of \(\mathcal{A}_U\) therefore cover \(U\). Any two charts of \(\mathcal{A}_U\) are smoothly compatible because they already belong to the smooth atlas \(\mathcal{A}\). So \(\mathcal{A}_U\) is a smooth atlas, and by the generation proposition it determines a unique smooth structure on \(U\).

The terminology "open submanifold" is one of the earliest instances of a substructure notion in differential geometry. A more general class of submanifolds (those that need not be open in the ambient manifold) requires substantially more machinery and will be taken up later in the manifold series. For now the open case is the only one available, and it is already enough to produce two of the most important examples in the subject.

The General Linear Group

The first such example is the general linear group of degree \(n\), the group of invertible \(n \times n\) real matrices under matrix multiplication.

The smooth structure on \(GL(n, \mathbb{R})\) is exactly the one inherited from \(M(n, \mathbb{R})\). A chart on \(GL(n, \mathbb{R})\) is smooth precisely when it is the restriction of a smooth chart on \(M(n, \mathbb{R})\). With respect to this structure, matrix multiplication \(GL(n, \mathbb{R}) \times GL(n, \mathbb{R}) \to GL(n, \mathbb{R})\) and matrix inversion \(GL(n, \mathbb{R}) \to GL(n, \mathbb{R})\) will be shown, later in the manifold series, to be smooth maps. The verification rests on the explicit polynomial formulas for the entries of a matrix product and the rational formulas (with nonvanishing denominator on \(GL(n, \mathbb{R})\)) for the entries of a matrix inverse. The same construction with \(\mathbb{C}\) in place of \(\mathbb{R}\) gives the complex general linear group \(GL(n, \mathbb{C}) \subseteq M(n, \mathbb{C})\), a smooth manifold of real dimension \(2n^2\).

Matrices of Full Rank

The general linear group is the special case \(m = n\) of a broader family. For a rectangular matrix \(A \in M(m \times n, \mathbb{R})\), the rank of \(A\) is the maximum number of linearly independent rows or columns, and it is at most \(\min(m, n)\). The matrix is said to have full rank when its rank attains this maximum.

The full-rank condition will recur throughout the manifold series and in applications. A linear layer of a neural network whose weight matrix is full-rank corresponds to an injective (or surjective) linear map. The columns of a full-column-rank matrix span a maximal-dimensional subspace. The assignment to each full-rank matrix of the subspace it spans is the first step in the construction of the Grassmannian manifold of \(m\)-planes in \(n\)-space, a smooth manifold that we will study in a later page of the manifold series.

Smooth Graphs

The graph construction encountered in the topological setting refines naturally to the smooth setting. Recall that for a continuous function \(f : U \to \mathbb{R}^k\) on an open subset \(U \subseteq \mathbb{R}^n\), the graph \[ \Gamma(f) = \{(x, f(x)) : x \in U\} \subseteq \mathbb{R}^n \times \mathbb{R}^k = \mathbb{R}^{n+k} \] is a topological \(n\)-manifold when equipped with the subspace topology inherited from \(\mathbb{R}^{n+k}\). Under this topology the restriction of the projection \(\pi_1 : \mathbb{R}^n \times \mathbb{R}^k \to \mathbb{R}^n\) onto the first factor gives a homeomorphism \(\varphi : \Gamma(f) \to U\), a single chart making \(\Gamma(f)\) a topological manifold. When \(f\) is smooth, this construction promotes to a smooth structure with no extra effort.

Suppose now that \(f : U \to \mathbb{R}^k\) is smooth, in the ordinary Euclidean sense. The single-chart atlas \(\{(\Gamma(f), \varphi)\}\) is, vacuously, a smooth atlas. With only one chart, the smooth compatibility condition is automatically satisfied. By the generation proposition, it determines a unique smooth structure on \(\Gamma(f)\), making the graph a smooth \(n\)-manifold. The chart \((\Gamma(f), \varphi)\) is, by definition, a smooth chart of this structure.

The smoothness of \(f\) is not used in proving that \(\Gamma(f)\) is a topological manifold. Continuity is enough for that. What the smoothness adds is the further claim that \(\Gamma(f)\) sits inside \(\mathbb{R}^{n+k}\) in a way that is itself compatible with the standard smooth structure on \(\mathbb{R}^{n+k}\). Read through the chart \(\varphi\), the inclusion map \(\Gamma(f) \hookrightarrow \mathbb{R}^{n+k}\) becomes the map \(x \mapsto (x, f(x))\) on \(U\), which is smooth as a map between Euclidean open sets. This compatibility is the smooth analogue of saying that \(\Gamma(f)\) is "smoothly embedded" in \(\mathbb{R}^{n+k}\), a notion that will be made precise when smooth submanifolds are taken up later in the series.