Differentiable structures

I am taking a course on differential topology this semester and will be writing a series of posts on this topic. In this post, we will look at the notion of differentiable structures on a topological manifold.

The foundational concepts of calculus, including differentiation, tangent vectors, and integration, are robustly defined within the familiar framework of Euclidean space, $\mathbb{R}^n$ . This setting provides a clear understanding of rates of change and accumulation. However, many spaces encountered in mathematics and physics, such as the surfaces of spheres or tori, are intrinsically curved or topologically complex and do not globally resemble $\mathbb{R}^n$ . The concept of a “manifold” emerges as a crucial abstraction, defining a topological space that locally “looks like” (is homeomorphic to) an open subset of $\mathbb{R}^n$ . This local resemblance allows for the extension of local coordinate systems from $\mathbb{R}^n$ to these more generalized spaces. The fundamental motivation for introducing differentiable structures on these topological manifolds is to enable the application of calculus on these generalized spaces. While a topological manifold provides the initial framework by ensuring local Euclidean resemblance through charts, it does not inherently provide the necessary structure for differentiation. For instance, the graph of $y= 2x+|x|$ a 1-dimensional topological manifold, but its sharp corner at the origin precludes differentiability at that point. The coordinates charts where you used $x$ coordinates and $y$ coordinates both define topological structure of the graph. The challenge then becomes how to ensure that local definitions of differentiability, imported from $\mathbb{R}^n$ via charts, remain consistent when different charts overlap.

This consistency is precisely what the differentiable structure provides. It is an additional layer of refinement imposed on a topological manifold, specifically by requiring that the mappings between different local coordinate systems (charts) are “smooth”. This ensures that calculus operations defined locally cohere into a global, well-defined notion of differentiability across the entire manifold. The term “differentiable manifold” often implicitly refers to a $C^{\infty}$ manifold, meaning infinitely differentiable, but the framework accommodates various degrees of smoothness, including $C^k$ (k-times continuously differentiable) and $C^{\omega}$ (real analytic).

Topological Manifold

To start with, n-manifolds are spaces which locally look like real vector space $\mathbb{R}^n$ . An n-dimensional topological manifold $M$ is formally defined as a topological space where every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset $V$ of $\mathbb{R}^n$ . This definition captures the intuitive idea that a manifold locally resembles Euclidean space.

More formally, a n-dimensional manifold $M^n$ is defined by the following structure:

1) A topological space $M=\bigcup_{\alpha}U_{\alpha}$ where $U_{\alpha}$ is collection of open sets covering $M$ .( Sometimes extra conditions of hausdorffness and second countability are imposed on the space. We don’t do it here.)

2) An atlas of charts $\phi_{\alpha}:U_{\alpha}\to V_{\alpha}$ which are homeomorphisms from $U_{\alpha}$ in $M$ to $V_{\alpha}$ in $\mathbb{R}^n$ s.

Using these charts we can identify the points in the space with points in $\mathbb{R}^n$ locally. However if we two different charts overlap same point in the space may be mapped to different points in $\mathbb{R}^n$ in different charts.

The concept of charts and atlases is the fundamental mathematical formalization of the intuitive idea that a manifold “locally looks like Euclidean space.” This construction allows for the translation of local properties from the familiar Euclidean setting to more general spaces. While the choice of specific charts or atlases is not unique for a given manifold, this local-to-global construction is fundamental to its definition. However, at this purely topological level, the “flatness” provided by the charts is only continuous, not differentiable. This limitation means that while one can define continuity on the manifold, it does not inherently allow for differentiation, necessitating the introduction of further structure for calculus. A topologically “nice” map might still correspond to a non-differentiable function in Euclidean space, such as a function with a cusp.

Note: The Hausdorff condition ensures that distinct points in a manifold can be topologically separated, avoiding pathological cases that hinder local or global analysis. Second countability requires a countable basis for the topology, enabling manageable topological structure. Together, these conditions guarantee that a manifold is paracompact and metrizable. Metrizability allows familiar notions like distance and convergence, while paracompactness ensures the existence of partitions of unity—essential tools for extending local constructions (like functions, vector fields, or metrics) to global ones. Without these properties, many core results and constructions in differential geometry would fail. Thus, Hausdorff and second countability are not technicalities—they’re essential for the theory to function reliably.

Differentiable Manifolds

Now we go to the differentiable structure on manifolds given which we can speak of differentiability, derivatives etc. n-dimensional differentiable structure (or $C^{k}$ -structure) on a topological manifold M is defined by a $C^{k}$ -atlas. That is a collection of coordinates charts $\phi_{\alpha}: V_{\alpha} \to U_{\alpha}$ which map subsets of $V_{\alpha} \subset\mathbb{R}^n$ to open subsets $U_{\alpha} \subset M$ .

Formally the differentiable structure is defined as follows

A function $f:U_{\alpha}\to \mathbb{R}$ is said to be differentiable if $f({\phi_\alpha}^{-1})$ is differentiable. That we require the function obtained by pulling back using $\phi_{\alpha}$ to be differentiable.(These are just functions from $\mathbb{R}$ to $\mathbb{R}.$ We know what differentiability of these functions mean.) So we now have a class of differentiable functions on $U_{\alpha},$ $D_\alpha =$ $\{f:U_{\alpha}\to \mathbb{R}$ $\mbox{ differentiable wrt to }\phi_\alpha \}$ .

Hence we have defined the local notion of differentiability (We can speak of functions being differentiable with respect to a particular chart). To speak of differentiability globally ( functions being differentiable on the manifold) we need these local notions to be consistent. For any $\alpha,\beta$ such that $U_{\alpha}\cap U_{\beta}\ne \phi$ we need that $D_{\alpha}| U_{\alpha}\cap U_{\beta}=D_{\beta}| U_{\alpha}\cap U_{\beta}$ . That is, we require every function on the intersection obtained by restriction, to be differentiable in both $\alpha$ notion and $\beta$ notion of differentiability or not differentiable in both. Said otherwise, we don’t want to have a function on the same region to be differentiable in $\alpha$ chart and not differentiable in the $\beta$ chart or vice versa. So thus imposing the consistency conditions we have a differentiable structure on our manifold.

Note that the compatibility of charts above is defined in terms of set of differentiable functions on the chart. But one can equivalently state the compatibility merely in terms of the charts $\phi_\alpha$ , $\phi_\beta$ . We require the transition functions $\phi_\alpha\circ\phi_\beta^-1$ and $\phi_\beta\circ\phi_\alpha^-1$ to be differentiable. It can be easily checked these notions are equivalent.

To summarise, differential structure is given by local notions which are turned into a global notion by imposing a compatibility through differentiable transition functions. One can similarly define the $C^k$ -differentiable structures, analytic structure on the manifold by requiring an atlas of compatible charts with transition functions being k times continuously differentiable, analytic respectively. One could also define similar structures with complex manifolds where the space looks like complex vector space locally. and we can talk of complex analytic structures.

Transition functions serve as the “coordinate change maps” that allow for translation between different local coordinate systems in regions where charts overlap. They explicitly describe how the coordinates of a point change when viewed through different charts. The differentiability of these transition functions is the fundamental property that ensures a consistent and well-defined notion of differentiability for functions on the manifold.

A maximal atlas for a given differentiable structure is the unique atlas that contains all possible $C^{k}$ compatible charts for that structure. It is formed by taking the union of all $C^{k}$ -atlases that are $C^{k}$ -compatible with each other. This maximal atlas uniquely determines the differentiable structure on the manifold. $C^{k}$ -differentiable manifold is formally defined as a topological manifold equipped with a maximal $C^{k}$ -atlas.

While the atlas-based definition is standard and intuitive, more abstract but powerful alternative approaches exist. These include defining differentiable structures via pseudogroups, which are collections of local diffeomorphisms that satisfy certain group-like properties , or through the theory of sheaves of local rings, which define algebras of functions on open sets of the manifold. These alternative viewpoints highlight the flexibility and generality of the underlying mathematical concepts, particularly useful for complex manifolds or in algebraic geometry. The concept of a maximal atlas, along with the existence of pseudogroups and sheaves, reveals that the “differentiable structure” is not merely an arbitrary collection of charts. Instead, it is a precisely defined mathematical object that encapsulates all possible consistent ways to view the manifold locally. This framework is remarkably flexible, allowing for the definition of not only smooth manifolds but also complex manifolds (requiring holomorphic transition functions), affine manifolds (requiring affine transition functions), and other geometric categories. This demonstrates a broader unifying principle in modern geometry, where the “smoothness” is just one particular choice of allowed local transformations

Calculus on Differentiable Manifolds

With a differentiable structure, one can rigorously define smooth functions on a manifold. A function $\displaystyle f: M \to \mathbb{R}$ is smooth if, for every chart $(U, \varphi)$ , the local expression $\displaystyle f \circ \varphi^{-1}: \varphi(U) \to \mathbb{R}$ is smooth in the Euclidean sense. Likewise, a map $\displaystyle f: M \to N$ between manifolds is smooth if, for all charts $(U, \varphi)$ on $M$ and $(V, \psi)$ on $N$ with $\displaystyle f(U) \subseteq V$ , the composition $\displaystyle \psi \circ f \circ \varphi^{-1}: \varphi(U) \to \psi(V)$ is smooth. Thanks to the atlas’s smooth compatibility, this definition is chart-independent.

The differentiable structure allows the definition of the tangent space $\displaystyle T_pM$ at each point $\displaystyle p \in M$ —an $n$ -dimensional real vector space representing the best linear approximation to $M$ at $p$ . Tangent vectors can be seen either as velocities of smooth curves through $p$ or as derivations on the algebra of smooth functions at $p$ . This structure enables the notion of direction, rate of change, and differentiation on curved spaces, all requiring consistent behavior across overlapping charts. A vector field on $\displaystyle M$ is a smooth assignment of a tangent vector to each point in $\displaystyle M$ . Vector fields generalize directional derivatives and form a Lie algebra under the Lie bracket, capturing the non-commutativity of flows. Dual to tangent vectors are cotangent spaces and differential forms, which allow for “measuring” and integration on manifolds. Differential forms generalize line, surface, and volume integrals and enable powerful results like Stokes’ theorem in a broad geometric setting.

From defining smooth functions and tangent vectors to vector fields, differential forms, and integration, the differentiable structure is the foundation enabling calculus on manifolds.

Examples

1. Euclidean Space ( $\displaystyle \mathbb{R}^n$ )

The archetypal differentiable manifold, $\displaystyle \mathbb{R}^n$ , serves as the local model for all other manifolds. Its differentiable structure is canonically defined by the identity map $\displaystyle (\mathbb{R}^n, \mathrm{id})$ , forming a trivial single-chart atlas. The absence of any overlapping chart domains means all compatibility conditions are vacuously satisfied. This tautological construction underscores the fundamental role of $\displaystyle \mathbb{R}^n$ as the space where our baseline notion of smoothness is established. Any open subset of $\displaystyle \mathbb{R}^n$ inherits this smooth structure directly.

2. The $\displaystyle n$ -Sphere ( $\displaystyle S^n$ )

The $\displaystyle n$ -sphere, defined as $\displaystyle S^n = {x \in \mathbb{R}^{n+1} : |x| = 1}$ , represents the simplest compact manifold that is not an open subset of $\displaystyle \mathbb{R}^n$ . Its standard differentiable structure is typically endowed via a minimal two-chart atlas derived from stereographic projection.

Let $\displaystyle N = (0, \dots, 0, 1)$ be the North Pole and $\displaystyle S = (0, \dots, 0, -1)$ be the South Pole.

North Pole Chart: $\displaystyle(U_N, \varphi_N)$ , where $\displaystyle U_N = S^n \setminus {N}$ , and

$\displaystyle \varphi_N(x_1, \dots, x_{n+1}) = \frac{1}{1 - x_{n+1}} (x_1, \dots, x_n)$

South Pole Chart: $\displaystyle (U_S, \varphi_S)$ , where $\displaystyle U_S = S^n \setminus {S}$ , and

$\displaystyle \varphi_S(x_1, \dots, x_{n+1}) = \frac{1}{1 + x_{n+1}} (x_1, \dots, x_n)$

The crucial transition function $\displaystyle \varphi_S \circ \varphi_N^{-1}$ maps $\displaystyle \mathbb{R}^n \setminus {0}$ to itself. For $\displaystyle y = (y_1, \dots, y_n) \in \mathbb{R}^n \setminus {0}$ , this map is given by

$\displaystyle y \mapsto \frac{|y|^2}{y}$ .

This is an inversion with respect to the unit sphere in $\displaystyle \mathbb{R}^n$ . As a composition of rational functions with non-vanishing denominators, it is $\displaystyle C^\infty$ .

Alternatively, $\displaystyle S^n$ can be covered by $\displaystyle 2(n+1)$ charts using projections onto coordinate hyperplanes. For $\displaystyle S^2$ , one can use six charts $\displaystyle (U_i^\pm, \varphi_i^\pm)$ where

$\displaystyle U_i^\pm = {(x_1, x_2, x_3) \in S^2 : \pm x_i > 0}$ .

For instance, $\displaystyle \varphi_3^+ (x_1, x_2, x_3) = (x_1, x_2)$
maps to the open unit disk in $\displaystyle \mathbb{R}^2$ . Transition functions involve identity maps, sign changes, and coordinate reorderings—all $\displaystyle C^\infty$ .

3. The $\displaystyle n$ -Torus ( $\displaystyle T^n$ )

The $\displaystyle n$ -torus is defined as $\displaystyle T^n = S^1 \times \cdots \times S^1$ ( $\displaystyle n$ times). It can be viewed as the quotient $\displaystyle \mathbb{R}^n / \mathbb{Z}^n$ .

A standard atlas uses open hypercubes in $\displaystyle \mathbb{R}^n$ slightly larger than 1 in side length,
e.g., $\displaystyle U_k = (k_1, k_1 + 1 + \varepsilon) \times \cdots \times (k_n, k_n + 1 + \varepsilon)$ for $\displaystyle k \in \mathbb{Z}^n$ , with charts $\displaystyle \varphi_k: U_k \to \mathbb{R}^n$ as restrictions of the projection $\displaystyle \pi: \mathbb{R}^n \to \mathbb{R}^n / \mathbb{Z}^n$ .

A common explicit construction uses $\displaystyle 2^n$ charts. For $\displaystyle T^2$ , four charts:

$\displaystyle U_1 = ((0,1) \times (0,1)) \bmod \mathbb{Z}^2$
$\displaystyle U_2 = ((0.5,1.5) \times (0,1)) \bmod \mathbb{Z}^2$
$\displaystyle U_3 = ((0,1) \times (0.5,1.5)) \bmod \mathbb{Z}^2$
$\displaystyle U_4 = ((0.5,1.5) \times (0.5,1.5)) \bmod \mathbb{Z}^2$

Each $\displaystyle U_i$ is homeomorphic to an open square in $\displaystyle \mathbb{R}^2$ . Transition maps involve shifts like $\displaystyle x \mapsto x + k$ , which are trivially $\displaystyle C^\infty$ . This construction reveals how a global identification leads to a natural smooth structure. It is notable that $\displaystyle T^n$ cannot be covered by fewer than three charts if those charts are required to have contractible domains, highlighting subtle topological constraints on atlas construction.

4. Real Projective Space ( $\displaystyle \mathbb{RP}^n$ )

Defined as the set of lines through the origin in $\displaystyle \mathbb{R}^{n+1}$ :
The points are represented by homogeneous coordinates.

$\displaystyle \mathbb{RP}^n = { [x_0 : \dots : x_n] \sim [\lambda x_0 : \dots : \lambda x_n] ,|, \lambda \in \mathbb{R} \setminus {0} }$

An atlas of $\displaystyle n+1$ charts is given by:

Chart domain: $\displaystyle U_i = {[x_0:\dots:x_n] : x_i \neq 0}$
Chart map:
$\displaystyle \varphi_i([x_0:\dots:x_n]) = \left(\frac{x_0}{x_i}, \dots, \widehat{\frac{x_i}{x_i}}, \dots, \frac{x_n}{x_i} \right)$

Transition functions (e.g., $\displaystyle \varphi_1 \circ \varphi_0^{-1}$ ) are rational functions.
Example:

$\displaystyle \varphi_1([1 : y_1 : \dots : y_n]) = \left( \frac{1}{y_1}, \frac{y_2}{y_1}, \dots, \frac{y_n}{y_1} \right)$ ,

which are $\displaystyle C^\infty$ where defined.

The projective spaces showcase how equivalence relations lead to compact manifolds whose smooth structures are defined by rational maps. $\displaystyle \mathbb{RP}^n$ is diffeomorphic to $\displaystyle S^1$ , a topologically distinct construction yielding a smoothly equivalent manifold.

5. Complex Projective Space ( $\displaystyle \mathbb{CP}^n$ )

Analogous to $\displaystyle \mathbb{RP}^n$ , is the set of one dimensional complex linear subspaces defined using complex homogenous coordinates:

$\displaystyle \mathbb{CP}^n = { [z_0 : \dots : z_n] \sim [\lambda z_0 : \dots : \lambda z_n] ,|, \lambda \in \mathbb{C} \setminus {0} }$

Charts:

Domain: $\displaystyle U_i = {[z_0:\dots:z_n] : z_i \neq 0}$
Map:
$\displaystyle \varphi_i([z_0:\dots:z_n]) = \left(\frac{z_0}{z_i}, \dots, \widehat{\frac{z_i}{z_i}}, \dots, \frac{z_n}{z_i} \right)$

Transition functions are holomorphic fractional linear maps, hence $\displaystyle C^\infty$ . Thus we even have a structure of a complex manifold (stronger than be a smooth manifold). One can show that $\displaystyle \mathbb{CP}^1 \cong S^2$ .

6. Lie Groups

Many fundamental examples of differentiable manifolds arise naturally as Lie groups. A Lie group $\displaystyle G$ is a differentiable manifold with group structure where multiplication $\displaystyle G \times G \to G$ and inversion $\displaystyle G \to G$ are smooth.

Examples:

$\displaystyle \mathrm{GL}(n, \mathbb{R}) = {A \in M(n, \mathbb{R}) : \det A \neq 0}$ , the set of invertible matrices. This is an open subset of $\displaystyle \mathbb{R}^{n^2}$ , hence it inherits a canonical smooth structure. Matrix multiplication and inversion (via adjugate formula) are smooth operations.
$\displaystyle \mathrm{O}(n) = {A \in \mathrm{GL}(n, \mathbb{R}) : A^T A = I}$ . This is a closed submanifold defined by polynomial equations. Its differentiable structure is derived from the implicit function theorem.
$\displaystyle \mathrm{SO}(n) = {A \in \mathrm{O}(n) : \det A = 1}$ . This is the connected component of the identity in $\displaystyle \mathrm{O}(n)$ inheriting its smooth structure.
$\displaystyle \mathrm{U}(n) = {A \in M(n, \mathbb{C}) : A^* A = I}$ . Similarly we can show these are these are complex manifolds, hence real smooth manifolds.

7. Submanifolds of $\displaystyle \mathbb{R}^N$

Many manifolds naturally arise as submanifolds of Euclidean space. A subset $\displaystyle M \subset \mathbb{R}^N$ is a smooth $\displaystyle n$ -dimensional submanifold if, for each $\displaystyle p \in M$ , there exists an open $\displaystyle U \subset \mathbb{R}^N$ and a diffeomorphism $\displaystyle \psi: U \to V \subset \mathbb{R}^N$ such that
$\displaystyle \psi(U \cap M) = V \cap (\mathbb{R}^n \times {0})$

Examples:

Open subsets of $\displaystyle \mathbb{R}^n$
$\displaystyle S^n \subset \mathbb{R}^{n+1}$
Graph of $\displaystyle f: \mathbb{R}^n \to \mathbb{R}^m$ :
$\displaystyle \Gamma_f = {(x, f(x)) : x \in \mathbb{R}^n}$
Embedded torus $\displaystyle T^2 \subset \mathbb{R}^3$

The Implicit Function Theorem ensures that level sets like $\displaystyle f(x) = |x|^2 - 1$ define smooth submanifolds such as $\displaystyle S^n$ .

8. Product Manifolds

Given two differentiable manifolds, $\displaystyle M$ of dimension $\displaystyle n$ and $\displaystyle N$ of dimension $\displaystyle m$ , their Cartesian product $\displaystyle M \times N$ naturally inherits a differentiable structure, forming an $\displaystyle (n+m)$ -dimensional differentiable manifold. This construction is fundamental for building higher-dimensional manifolds from simpler ones (e.g., tori from circles, cylinders from circles and lines).

Let $\displaystyle {(U_\alpha, \varphi_\alpha)}$ be a $\displaystyle C^k$ -atlas for $\displaystyle M$ , and $\displaystyle {(V_\beta, \psi_\beta)}$ be a $\displaystyle C^k$ -atlas for $\displaystyle N$ .

Charts for $\displaystyle M \times N$ : The product atlas consists of charts of the form $\displaystyle (U_\alpha \times V_\beta, \chi_{\alpha\beta})$ , where

$\displaystyle \chi_{\alpha\beta}: U_\alpha \times V_\beta \to \varphi_\alpha(U_\alpha) \times \psi_\beta(V_\beta) \subset \mathbb{R}^n \times \mathbb{R}^m \cong \mathbb{R}^{n+m}$

is given by $\displaystyle \chi_{\alpha\beta}(p,q) = (\varphi_\alpha(p), \psi_\beta(q))$ .

Transition Functions: Consider two such charts. Their transition function is:

$\displaystyle T = \chi_{\alpha'\beta'} \circ \chi_{\alpha\beta}^{-1} : (\varphi_\alpha(U_\alpha \cap U_{\alpha'}) \times \psi_\beta(V_\beta \cap V_{\beta'})) \to (\varphi_{\alpha'}(U_\alpha \cap U_{\alpha'}) \times \psi_{\beta'}(V_\beta \cap V_{\beta'}))$

Explicitly, $\displaystyle T(x,y) = (\varphi_{\alpha'} \circ \varphi_\alpha^{-1}(x), \psi_{\beta'} \circ \psi_\beta^{-1}(y))$ .

Since the components are $\displaystyle C^k$ , their product is $\displaystyle C^k$ as well.

9. Graphs of Smooth Functions

Let $\displaystyle f: U \to \mathbb{R}^m$ be a $\displaystyle C^k$ function, where $\displaystyle U \subset \mathbb{R}^n$ is open. The graph of $\displaystyle f$ is

$\displaystyle \Gamma_f = {(x, f(x)) \in \mathbb{R}^{n+m} : x \in U}$ ,

an $\displaystyle n$ -dimensional smooth manifold.

Define $\displaystyle \psi: \Gamma_f \to U \subset \mathbb{R}^n$ by $\displaystyle \psi(x, f(x)) = x$ , a homeomorphism with inverse $\displaystyle \psi^{-1}(x) = (x, f(x))$ .

Smoothness follows from the smoothness of $\displaystyle f$ . Any coordinate change involves components of $\displaystyle f$ and its inverse, which are $\displaystyle C^k$ .

10. Manifolds by Gluing: The Möbius Strip

Not all manifolds are easily defined as subsets of Euclidean space or products. Some are constructed by identifying boundaries of simpler pieces. The Möbius strip is a quintessential example of such a construction.

Define $\displaystyle M = ([0,1] \times (-1,1)) / \sim$ with $\displaystyle (0,y) \sim (1, -y)$ .

Let $\displaystyle \pi: [0,1] \times (-1,1) \to M$ be the quotient map.

Chart 1: Let $\displaystyle W_1 = (1/4, 3/4) \times (-1,1)$ , then $\displaystyle U_1 = \pi(W_1)$ and $\displaystyle \varphi_1 = \pi|_{W_1}^{-1}$ .

Chart 2: Let
$\displaystyle W_2 = ((-1/4,1/4) \cup (3/4,5/4)) \times (-1,1)$ ,
then $\displaystyle U_2 = \pi(W_2)$ and $\displaystyle \varphi_2$ maps:

$\displaystyle (x,y) \mapsto (x,y)$ for $\displaystyle x \in (-1/4,1/4)$
$\displaystyle (x,y) \mapsto (x-1, -y)$ for $\displaystyle x \in (3/4, 5/4)$

On overlaps, transitions are $\displaystyle C^\infty$ (identity or affine flips).

11. Tangent Bundle ( $\displaystyle TM$ )

This is a crucial example as its smooth structure is derived directly from that of M, and it is fundamental for defining vector fields, tensor fields, and differential forms.

Let $\displaystyle (U_\alpha, \varphi_\alpha)$ be a $\displaystyle C^\infty$ chart on $\displaystyle M$ . Define $\displaystyle \Phi_\alpha: \pi^{-1}(U_\alpha) \to \varphi_\alpha(U_\alpha) \times \mathbb{R}^n$ by

$\displaystyle \Phi_\alpha(p, v) = (\varphi_\alpha(p), (\varphi_\alpha)_* v)$

Transition maps between charts are:

$\displaystyle T_{\beta\alpha}^{TM}(x, w) = ((\varphi_\beta \circ \varphi_\alpha^{-1})(x), D(\varphi_\beta \circ \varphi_\alpha^{-1})(x) \cdot w)$

These are $\displaystyle C^\infty$ since derivatives of smooth maps are smooth.

12. Quotient Manifolds by Free and Proper Group Actions

If a Lie group $\displaystyle G$ acts freely and properly on $\displaystyle X$ , then the quotient $\displaystyle X/G$ inherits a smooth structure.

Examples:

$\displaystyle T^n = \mathbb{R}^n / \mathbb{Z}^n$
$\displaystyle \mathbb{RP}^n = S^n / {\pm I}$
Klein Bottle: $\displaystyle \mathbb{R}^2 / \Gamma$ where $\displaystyle \Gamma$ is generated by $\displaystyle (x,y)\mapsto (x+1,y)$ and $(x,y)\mapsto(x,1-y)$ .

Transition functions on the quotient are smooth, inherited from $\displaystyle X$ and the group action.

Existence and Uniqueness of differentiable structures

We have a defined what we mean by a differentiable structure. Now we can ask if every topological manifold admits a differentiable structure. If it does, is it unique? If there are more, how different are these structures? If we decide to consider some types of structures to be equivalent, we can ask how many different structures exist when we quotient out the equivalences.

The existence and uniqueness of differentiable structures on topological manifolds are among the most involved questions in differential topology. The answers are far from trivial and depend critically on the dimension of the manifold.

Topological Manifolds Without Differentiable Structures (Non-Smoothable Manifolds)

A fundamental discovery in differential topology is that not every topological manifold can be endowed with a differentiable structure. This means that being “locally Euclidean” (a topological manifold) is a strictly weaker condition than being “locally smoothly Euclidean” (a differentiable manifold). This negative answer to the existence question is profound, as it implies that the topological notion of “locally Euclidean” is strictly weaker than the differentiable notion. There are spaces that are topologically well-behaved but inherently incompatible with the requirements of a global smooth structure. The existence of such manifolds suggests that the local compatibility conditions for differentiability have deep and non-trivial global topological consequences, restricting which topological spaces can support calculus.

A celebrated example is the E8 manifold, a 4-dimensional topological manifold that definitively does not admit any differentiable structure. This striking discovery highlights that some spaces, while topologically “nice,” fundamentally resist being “smoothed out” everywhere in a consistent manner.
The E8 manifold demonstrably admits no differentiable structure whatsoever. This striking result highlights a fundamental divergence between the categories of topological and smooth manifolds in dimension 4. Beyond dimension 4, M. Kervaire demonstrated in 1960 the existence of a compact triangulable 10-dimensional manifold that is also non-smoothable. These examples demonstrate that the “smooth” category of manifolds is a proper subcategory of the “topological” category. Not all topological spaces that are locally Euclidean can support a consistent global calculus. This reveals a fundamental difference in the properties that “smoothness” imposes compared to mere “continuity,” implying that the local differentiability constraints on transition functions have profound global topological repercussions, preventing certain topological manifolds from ever being “smoothed.”

“Exotic” Differentiable Structures (Non-Unique Structures)

Even if a topological manifold does admit a differentiable structure, that structure might not be unique. Different maximal atlases can exist on the same topological manifold, leading to distinct “exotic” differentiable structures. These structures are topologically equivalent (homeomorphic) but are not smoothly equivalent (not diffeomorphic). This implies that topology alone is insufficient to classify manifolds up to smooth equivalence. Differential geometry, by imposing the differentiable structure, uncovers finer, deeper distinctions between spaces that appear identical from a purely topological standpoint.

The most famous discovery in this area is John Milnor’s 1956 construction of exotic 7-spheres. These are manifolds homeomorphic to the standard but not diffeomorphic to it. Milnor’s work, which earned him a Fields Medal, disproved the differentiable Poincaré conjecture and revealed that $\displaystyle S^7$ admits at least 28 distinct differentiable structures. This finding revolutionized geometric topology, demonstrating that objects indistinguishable at a topological level can be profoundly distinct when endowed with a smooth structure. ilnor constructed these exotic spheres as total spaces of certain $\displaystyle 3 0$ -sphere bundles over the $\displaystyle 4 0$ -sphere. Using Morse theory, he showed that they are homeomorphic to $\displaystyle S^7 0$ , but not diffeomorphic to $\displaystyle S^7 0$ . To distinguish them, he introduced an invariant of $\displaystyle 7 0$ -manifolds that is preserved under diffeomorphism but not under homeomorphism.

The landscape of uniqueness is highly dimension-dependent:

Dimensions 1, 2, 3: In these low dimensions, every topological manifold admits a unique differentiable structure (up to diffeomorphism). This is a result of Radó (1925) for n=1,2 and Moise (1952) for n=3, simplifying the classification problem considerably.
Dimension 4: This dimension is uniquely complex and remains a frontier of active research. It is known that $\mathbb{R}^4$ itself admits uncountably many distinct differentiable structures, a phenomenon known as exotic R4. For compact 4-manifolds, the situation is similarly intricate, with classification relying on powerful invariants like Donaldson and Seiberg-Witten invariants, highlighting the deep interplay with gauge theory.
Dimensions $\ge 5$ : For $n\ge 5$ , all topological n-spheres $\displaystyle S^n$ are smoothable. However, they can admit multiple distinct smooth structures, as shown by Milnor’s work.

Differentiable Manifolds

1. Euclidean Space ()

2. The -Sphere ()

3. The -Torus ()

4. Real Projective Space ()

5. Complex Projective Space ()