ABSTRACT
In this three-part series, we rigorously explore the concept of manifolds through the perspectives of both differential geometry and discrete differential geometry. In Part I, we establish the formal definition of a manifold as a special type of topological space and present illustrative examples. In Part II, we introduce the additional structure needed to define differentiable manifolds. Finally, part III presents the discretization of manifolds within the framework of discrete differential geometry, where we approximate smooth manifolds using simplicial complexes or polygonal meshes. Looking at the concept from both perspectives is an opportunity to gain a deeper insight into both types of geometries. The series is nearly self-contained, requiring only a basic understanding of naive set theory and elementary calculus from the reader.
Introduction
A manifold is a special kind of topological space, so special, in fact, that mathematicians have given it its own name. The term “manifold” traces back to the Old English manigfeald and Proto-Germanic maniġfaldaz, meaning “many folds” or “layers.” This etymology descriptively captures the essence of what a manifold represents: a space with many dimensions or complexities, yet with a coherent structure. To define a manifold formally, we first introduce the concept of a general topological space. Only after this, we can talk about the specific properties that a topological space must have to be considered a manifold.
Topological Spaces
Definition. Let \( M \) be a set. Then a choice \( \mathcal{O} \subseteq \mathcal{P}(M) \) is called a topology on \( M \) if:
- \( \emptyset \in \mathcal{O} \) and \( M \in \mathcal{O} \);
- For \(\{U_i\}_{i=1}^n \subseteq \mathcal{O}\) \( \Rightarrow \bigcap \{U_i \}_{i=1}^n \in \mathcal{O}\)
- For any arbitrary collection of sets \( \mathcal{C} \subseteq \mathcal{O}\) \( \Rightarrow \bigcup \mathcal{C} \in \mathcal{O}\)
And the pair \( (M, \mathcal{O}) \) is called a topological space.
Abuse of Notation. In this note, sometimes we abbreviate \(M, \mathcal{O}\) by just \(M\), leaving the topology \( \mathcal{O}\)
implicit.
In mathematics, a topology on a set provides the weakest structure needed to define the two very important notions of convergence of sequences to points in a set, and of continuity of maps between two sets. Unless \( |M|=1 \). There are many different topologies one could establish on a set on the same set. Depending on what topology you have on \(M\), the notion of continuity and convergence changes accordingly.
The following table shows us how many different topologies one can establish on a set based on its cardinality.
| \( |M| \) | Number of Topologies |
| 1 | 1 |
| 2 | 4 |
| 3 | 29 |
| 4 | 355 |
| 5 | 6,942 |
| 6 | 209,527 |
| 7 | 9,535,241 |
Examples of Topologies
- Chaotic (trivial) topology: For the set \( M = \{a, b, c\} \), the chaotic topology includes only the entire set and the empty set: \( \mathcal{O} = \{\emptyset, M\} \). This topology is called “chaotic” because it has the least structure, and can be defined on any set.
- Discrete Topology: For the set \( M = \{a, b, c\} \), the discrete topology includes every possible subset of \( M \): \( \mathcal{O} = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\},\{b, c\}, \{a, b, c\}\}\). This topology provides the most structure, and can be defined on any set.
- Standard Topology on \( \mathbb{R} \) (Open Interval Topology): For the set \( M = \mathbb{R} \) (the real numbers), the standard topology is generated by open intervals \( (a, b) \) where \( a, b \in \mathbb{R} \) and \( a < b \): \( \mathcal{O} = \{U \subseteq \mathbb{R} \mid U \text{ is a union of open intervals } (a, b)\}\)
Just as sets are distinguished from each other based on one important property—the cardinality of sets—in set theory, we can define properties that help distinguish one topological space from another. There are many such topological properties for this purpose. We will present those needed to distinguish a topological space that is a manifold from one that is not, namely, the separation, compactness, and paracompactness properties.
Separation, Compactness, and Paracompactness of Topological Spaces
Separation Properties:
Separation properties are used to distinguish points and sets within a topological space, providing a way to understand how “separate” or “distinct” different points or subsets are. To illustrate, consider
\(M = \{a, b, c\}\) and the topology \( \mathcal{O} = \{\phi, \{a, b, c\}\}\). This topology is fairly “blind to its element”: it can not tell apart any of the points \(a, b, c\)! But any metric space can tell its points apart
(because \(d(x, y) > 0 \) when \( x \neq y\)). While we focus on one specific type of separation property—the \(T_2\) Hausdorff property—there are many other separation properties (many \(Ts\)), some stronger while others are weaker than \(T_2\), that also play important roles in topology.
Definition: A topological space \( (M, \mathcal{O}) \) is called a Hausdorff space (or \(T_2\) space) if for any two distinct points \( p, q \in O \), there exist disjoint open neighborhoods \( U \) and \( V \) such that \( p \in U \) and \( q \in V \). That is, the space satisfies the following condition:
For any \( p, q \in \mathcal{O}\) with \(p \neq q,\) there exist disjoint open sets \(U\) and \(V\) such that \(p \in U\) and \(q \in V. \)
Example: Consider the topological space \( (\mathbb{R}^2, \mathcal{O}) \), where \( \mathcal{O} \) is the standard topology on \( \mathbb{R}^2 \). This space is \(T_2\) Hausdorff. The standard topology \( \mathcal{O} \) on \( \mathbb{R}^2 \) is the collection of all unions of open balls.
An open ball centered at a point \( (x_0, y_0) \) with radius \( r > 0 \) is:
\( B((x_0, y_0), r) = \{ (x, y) \in \mathbb{R}^2 \mid \sqrt{(x – x_0)^2 + (y – y_0)^2} < r \} \)
And indeed, \( \mathbb{R}^2 \) has the\(T_2\) (Hausdorff) property since given any two distinct points in \( \mathbb{R}^2 \), you can always find two open balls that do not overlap.
More generally, the topological space \( (\mathbb{R}^d, \mathcal{O} )\) is \(T_2\) Hausdorff where \( \mathcal{O} \) is its standard topology.
Compactness, and Paracompactness:
Definition. Let \( (M, \mathcal{O}) \) be a topological space. An open cover of \( M \) is an arbitrary collection of open sets \( \{ U_{\alpha \in A} \}\) from \( \mathcal{O}\) (possibly infinite or finite) such that: \[ M = \bigcup_{\alpha \in A} U_{\alpha} \]
A subcover is exactly what it sounds like: it takes only some of the \(U_{\alpha \in A}\), while ensuring that \(M\) remains covered.
Definition. A topological space ( \(M, \mathcal{O}\) ) is called compact if every open cover of \( M \) has a finite subcover (i.e. there exists \( F \subset A \) such that: \(M = \bigcup_{\alpha \in F} U_{\alpha}\) where \(F\) is finite).
Compactness is a property that generalizes the notion of closed and bounded sets in Euclidean space. A topological space is compact if every open cover of the space has a finite subcover. This means that, no matter how the space is covered by open sets, it is possible to select a finite number of those sets that still cover the entire space. Compact spaces have several important properties.
In many mathematical contexts, when developing and proving new theorems within the framework of topological spaces, it is common to first address the case where the space is compact. Once the theorem/proof is established for compact spaces, efforts are then made to extend the result to non-compact spaces. Sometimes it is not possible to do the extension. On the other hand, paracompactness is a generalization of compactness (i.e, a much weaker notion) and rarely is it the case to find a topological space that is not paracompact.
Paracompactness:
Definition. A topological space \( (M, \mathcal{O}) \) is called paracompact if every open cover has an open refinement that is locally finite.
Given an open cover \( \{ U_{\alpha \in A} \}\) of \(M\), an open refinement \( { V_{\beta} }_{\beta \in B} \) of this cover is another open cover where every \( V_{\beta} \) is contained in some \( U_{\alpha} \) (i.e. \(\{ V_{\beta} \}_{\beta \in B}\) is a refinement if \( V_{\beta} \subset U_{\alpha} \text{ for some } \alpha \in A.\))
In other words, \( { V_{\beta} }_{\beta \in B} \) is a finer cover than \( \{ U_{\alpha \in A} \}\), meaning that each open set in the refinement is more “localized” or “smaller” in some sense compared to the original cover.
Definition. The refinement is said to be locally finite if every point in \( M \) has a neighborhood that intersects only finitely many of the sets \( V_{\beta} \).
This means that around any given point, only a finite number of the open sets in the cover are “active” or have non-empty intersections with the neighborhood.
In summary: Compactness ensures that any cover can be reduced to a finite cover, while paracompactness ensures that any cover can be refined to a locally finite cover. Compactness deals with the ability to reduce the size of a cover, while paracompactness deals with the ability to organize the cover more effectively without too much local overlap.
Now, we are ready to lay down the formal definition of a manifold!
Manifolds
Definition: A paracompact, Hausdorff topological space ( \(M, \mathcal{O}\) ) is called a (d)-dimensional manifold if for every point \( p \in M \), there exists a neighborhood \( U(p) \) of \( p \) and a homeomorphism \( \varphi: U(p) \to \varphi(U(p) ) \subset \mathbb{R}^d \). In this case, we also write dim \(M\)= \( d \).
What are homeomorphisms? Homeomorphism (Homeos) are structure-preserving maps between topological spaces. Formally, we say that a map \( \varphi: (M, \mathcal{O}_M) \to (N, \mathcal{O}_N) \) is called a homeomorphism if it satisfies the following conditions:
- \( \varphi: (M, \mathcal{O}_M) \to (N, \mathcal{O}_N) \) is a bijection
- \( \varphi: (M, \mathcal{O}_M) \to (N, \mathcal{O}_N) \) is continuous
- The inverse map \( \varphi^{-1}: (N, \mathcal{O}_N) \to (M, \mathcal{O}_M) \) is also continuous.
This definition tells us that a d-manifold is a special type of a topological space where we can distinguish between its subspaces, and it gives us two equivalent ways to think about it:
- Locally: for any arbitrary point \(p \in M\), you can always find an open set that contains it and this open set can be mapped by some homeo to a subset of \(\mathbb{R}^d\). For example, to someone standing on the surface of the Earth, the Earth looks much like \(\mathbb{R}^2\).
- Globally: there exists an open cover \( \{ U_{\alpha \in A} \}\) (possibly infinite) of \(M\) such that every \(U_{\alpha}\) is mapped by some homeo to a subset of \(\mathbb{R}^d\). For example, from outer space, the Earth can be covered by two hemispherical pancakes.
Examples of Manifolds
- The sphere \(S^2\) is a 2-manifold: every point in the sphere has a small open neighborhood that looks like a subset of \(\mathbb{R}^2\). One can cover the Earth with just two hemispheres, and each hemisphere is homeomorphic to a disk in \(\mathbb{R}^2\).
- The circle \(S^1\) is a 1-manifold; every point has an open neighborhood that looks like an open interval.
- The torus \(T^2\), and Klein bottle are 2-manifold too.
A non-example of a topological space that is not a manifold is the \(n\)-dimensional disk \(D^n\), because it has a boundary; points on the boundary do not have open neighborhoods that can be mapped by some homeo to a subset of \(\mathbb{R}^n\).
Definition. The closed n-dimensional disk, denoted by \( D^n \), is defined as the set of all points \( \mathbf{x} \in \mathbb{R}^n \) such that the Euclidean norm of \( \mathbf{x} \) is less than or equal to 1. Formally,
\[ D^n = \{ \mathbf{x} \in \mathbb{R}^n \mid |\mathbf{x}| \leq 1 \} \]
where \( |\mathbf{x}| = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2} \) is the Euclidean norm of the vector \( \mathbf{x} = (x_1, x_2, \dots, x_n) \).
Additional terminology: Atlases and Charts
The Terminology of a Chart on a \(d\)–manifold:
Let \(M\) be a \(d\)-manifold then, a chart on \(M\) is a pair \( (U, \varphi) \), where:
- \( U \) is an open subset of \( M \).
- \( \varphi: U \to \varphi(U) \subset \mathbb{R}^d \) (often called the coordinate map or coordinate chart) is a homeomorphism.
The component functions of \( \varphi: U \to \varphi(U) \subset \mathbb{R}^d \) are the mappings:
\[\varphi^{i}: U \to \mathbb{R}\]
\[p \mapsto proj_i(\varphi(p))\]
For \(1 \leq i \leq d\), where \(proj_i(\varphi(p))\) is the \(i\)-th component of \( \varphi (p) \in \mathbb{R}^d\).
This means that the map \( \varphi \) takes every point \(p\) in \( U \) and assigns it coordinates \(proj_i(\varphi(p))\) in \( \mathbb{R}^d = \mathbb{R}\times \mathbb{R} \times \dots \times \mathbb{R}\) ) ( \(d\) times) with respect to the chart \((U, \varphi) \).
Remarks.
- Notice that the paragraph above does not introduce any new information beyond what is contained in the definition of a \(d\)-topological manifold. This is why a “chart” is more of a terminology than a definition—though it is a useful one.
- We can see by now that there can exist a set \( \mathscr{A}\) of charts for each open set in the open cover \( \{ U_{\alpha \in A} \}\) of \(M\), and there will be many charts that overlap because Different charts may be needed to cover the entire manifold because a single chart might not be able to cover the entire surface of a sphere without singularities or overlaps.
Definition. An atlas of a manifold \( M \) is a collection \( \mathscr{A} := \{(U_\alpha, \varphi_\alpha) \mid \alpha \in A\} \) of charts such that:\[\bigcup_{\alpha \in A} U_\alpha = M.\]
Well, where do you think the words “chart” and “atlas” come from? 🙂
So what happens then if charts overlap? A natural map called the transition map displays itself naturally and is always continuous as a result of the original definition of the topological \(d\)-manifold.
Definition. Two charts \((U_1, \varphi_1)\) and \((U_2, \varphi_2)\) are called \(C^0\)-compatible if either:
- \(U_1 \bigcap U_2 \neq \phi \)
- \(U_1 \bigcap U_2 = \phi \): the (transition) map \( \varphi_2 \circ \varphi_1^{-1} : \varphi_1(U_1 \bigcap U_2) \to \varphi_2(U_1 \bigcap U_2)\) is continuous
By definition, one can go from \(U_1\) into \(\varphi_1 (U_1) \subseteq \mathbb{R}^d\), and similarly one can go from \(U_2\) into \(\varphi_2 (U_2) \subseteq \mathbb{R}^d\). For all the points in the \( U_1 \cap U_2 \), one could use either apply \(\varphi_1\) or \( \varphi_2 \) to land in the subsets \( \varphi_1 (U_1 \cap U_2) \) or \( \varphi_2 (U_1 \cap U_2) \) of \( \mathbb{R}^d \). All of a sudden, we constructed a map that goes from \( \varphi_2 \circ \varphi_1^{-1}: \mathbb{R}^d \to \mathbb{R}^d\) and this map is always continuous
This definition seems redundant and this is true, it applies to every pair of charts. However, it is just a “warm up” since we will later refine this definition and define the differentiability of maps on a manifold in terms of \(C^k\)-compatibility of charts.
Example. Consider a 2-dimensional manifold ( M ), such as the surface of a globe (a sphere). One chart ( (U_1, \varphi_1) ) might cover the Northern Hemisphere, with ( \varphi_1 ) assigning each point in ( U_1 ) latitude and longitude coordinates. Another chart ( (U_2, \varphi_2) ) might cover the Southern Hemisphere. In the overlap ( U_1 \cap U_2 ), the transition map ( \varphi_2 \circ \varphi_1^{-1} ) converts coordinates from the Northern Hemisphere chart to the Southern Hemisphere chart.
Remark. The structure of a topological \(d\)-manifold \(M\) allows us to distinguish subspaces (sub-manifolds) from each other and provides the framework to discuss the continuity of functions defined on \(M\). For example, if you have a curve \( c: \mathbb{R} \to M\) on the manifold, a function \( \mu: \mathbb{R} \to M \) or even a map \(\phi: M \to M\) you can talk about the continuity of \(c\), \(\mu\) and \(\phi\). However, the topological structure alone is not sufficient to discuss their differentiability. To do so, we need to impose an additional structure on \(M\), such as a smooth structure, to define and talk about differentiability.
In part II, we will talk more about Differentiable Manifolds.
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