Chapter 1: The Concept of Manifolds

In the realm of mathematics, Euclidean space serves as the foundational context for Calculus. However, the introduction of manifolds enables the application of Calculus to spaces that are not strictly flat or Euclidean.

Manifolds have been a part of our discussions since the beginning of this series, although we only explicitly identified them in our piece on Differential Geometry. At that point, we indicated that a manifold must exhibit local Euclidean characteristics; however, this is merely one of three essential traits we seek in manifolds.

More precisely, we aim for a manifold to be a mathematical structure that is locally Euclidean, possesses unique limits, and allows for integration.

If a space meets all these criteria, it qualifies as a manifold.

Check Your Understanding

For this section, there isn't much to do since I aim to clarify the rationale behind the definitions of manifolds. If you seek more questions, consider researching the bolded terms within this article and explore the linked resources in the Further Reading section.

Spaces That Do Not Qualify as Manifolds

The following examples illustrate spaces that do not fulfill manifold criteria:

• A figure-eight shape
• A cone
• The Cantor set
• The combination of a line and a filled circle intersecting
• The particular point topology, where open sets include all sets containing point p
• The Rational Numbers
• A sphere with hair

Partitions of Unity

To demonstrate that integration on a manifold is independent of the chosen partition of unity, we need to establish certain mathematical foundations.

Existence of Riemannian Metrics on Arbitrary Smooth Manifolds

Utilizing partitions of unity, second-countability, and the Hausdorff condition, we can prove that every smooth manifold is endowed with a Riemannian metric.

Prerequisites

This article includes numerous illustrations, minimizing the necessity for complex calculations. We have previously discussed local Euclidean properties in our Differential Geometry introduction, and much of the topology has been addressed in our Topology article. If you grasp the essence of limits, you will be adequately prepared for this discussion.

Recap

The first portion of this recap will revisit concepts from Differential Geometry, while the second will touch on Topology.

Euclidean Spaces

A Euclidean space is denoted as ℝⁿ, possessing notable characteristics:

• Parallel lines maintain their distance.
• The Pythagorean Theorem, (a² + b² = c²), holds true.
• The angles in a triangle sum to 180° or π radians.
• Movement in any direction increases distance from the starting point.
• The shortest distance between two points is represented by a straight line.

Euclidean space is particularly advantageous for Calculus due to its simplicity.

Non-Euclidean Spaces

A non-Euclidean space deviates from ℝⁿ. In contrast to Euclidean spaces, non-Euclidean spaces may lack several favorable properties:

• Parallel lines can intersect.
• The Pythagorean Theorem may not apply.
• The angles of a triangle might not add to 180°.
• Traveling in any direction can lead you back to your starting point (as with a sphere) or near it (like on a flat torus with an irrational velocity ratio).

In many instances, the nature of the space prevents the establishment of a vector space, necessitating adjustments in our Calculus approach.

Locally Euclidean Spaces

Often, we can mitigate the complications by examining small regions within a non-Euclidean space. For instance, the Earth's surface appears relatively flat within the confines of a city. Additionally, the Pythagorean Theorem has a minimal error margin of 0.002% at this scale. In urban environments:

• Parallel lines remain equidistant.
• Triangle angles approximate 180°.
• Traveling in one direction does not lead you back into the city from the opposite end.
• The shortest distance between points is nearly a straight line.

A space that can be approximated by a Euclidean space at all points is termed a locally Euclidean space.

Neighborhoods and Open Sets

A neighborhood consists of points that are "close" to a specific point. The definition of "close" varies based on the topological space in question, but certain characteristics about neighborhoods are consistent. To avoid complications, we generally work with open sets.

An open set is defined as a neighborhood encompassing every point within it. When dealing with open sets, three rules apply:

1. The empty set ∅ and the entire set X are open.
2. The union of open sets is open.
3. The intersection of a finite number of open sets is open.

These principles simplify our work, although they may sacrifice some intuitive clarity.

Charts and Atlases

Our initial task is to comprehend what it means for a space to be locally Euclidean.

Locally X

The term "locally X" indicates the existence of an open neighborhood around any chosen point where the property X applies, regardless of the point selected. For manifolds, this implies examining contiguous areas rather than the entire manifold simultaneously.

Locally Euclidean

To be locally Euclidean, we must identify contiguous regions in which the manifold approximates Euclidean characteristics. This entails taking a segment of the manifold, "flattening" it, and engaging in Euclidean operations.

In early Differential Geometry, we often utilized maps of the globe, referring to these regions as charts (also known as coordinate charts, coordinate patches, or local frames).

The Formal Definition of a Chart

A chart constitutes a mapping from an open subset of a manifold to an open subset of a Euclidean space. Here, "open" pertains to the open sets discussed in topology, meaning a chart specifies the region it flattens and the method of flattening.

Atlases

To describe a manifold comprehensively, we must cover every point with at least one chart. A collection of charts that entirely covers the manifold constitutes an atlas.

Overlapping Charts

While it is possible to create charts that do not overlap, doing so can be cumbersome. Instead, we allow overlapping charts, provided they do not cause contradictions. If two charts share coverage over the same point, there must exist a way to transition between them, known as a transition map.

Transition maps enforce the requirement that overlapping charts provide a consistent description of the space in their shared regions.

Differentiable, Smooth, and Analytic Manifolds

In principle, transition maps can take various forms, but we typically prefer some constraints. If our transition maps maintain derivatives everywhere, we classify the manifold as differentiable. If we can derive the transition maps repeatedly, it is termed a smooth manifold. An analytic manifold is one where the transition maps are analytic. For the purpose of integration on a manifold, we primarily require smooth manifolds.

Limits

To perform Calculus on manifolds, we need well-defined limits. This section presumes a basic understanding of limits, but additional resources are available if clarification is needed.

The Line With Two Origins

To ensure well-defined limits, we should examine a space where limits fail. We will consider the line with two origins, defined as T₁, where specific points p and q serve as unique representations of 0.

If we modify a continuous function to operate on this line, we can encounter issues with limits, as they can lead to ambiguity between p and q.

Uniqueness

When a result is singular, it is termed unique. For instance, there is a unique solution to (x + 3 = 8) within the integers. Conversely, if multiple results arise without a means to select one, we state that uniqueness is absent.

Uniqueness is critical for limits; if limits are not unique, we cannot guarantee unique solutions for differential equations, which undermines the predictability of outcomes based on those equations.

Hausdorff Spaces

The line with two origins fails to provide unique limits as outputs approach the origins because overlapping open sets containing p also overlap with those containing q. To avoid this issue, we need to establish that we can consistently identify non-overlapping open sets containing one point while excluding the other. If this condition is satisfied, the space is termed Hausdorff.

Partitions of Unity

Limiting ourselves to locally Euclidean and Hausdorff spaces results in topological manifolds. While topological manifolds are intriguing, they do not offer sufficient guarantees for integration.

What We Can Calculate

Integrals, as defined, apply solely to regions in Euclidean space, necessitating the division of regions into parallelotopes defined by vectors for each dimension.

To facilitate these calculations, we must operate within a vector space—practically, within Euclidean space. Given at least topological manifolds, we can map any chart of the space and the corresponding function into Euclidean space for integration. By aggregating results over the entire space, we approximate the integral.

The process can be visualized by considering each quadrilateral as a chart once flattened. Smaller charts yield more accurate approximations (e.g., a city map versus a global map).

Intersecting Charts

While practical integrations over manifolds typically involve non-overlapping charts, it is beneficial to allow intersections to simplify atlas definitions. However, this raises concerns about over-counting contributions in overlapping regions.

An Example

To illustrate the risk of over-counting, consider integrating a function (f(θ)) over a circle. If we define four charts for the integration, we might double our expected total charge since most points are covered by multiple charts.

To correct this, we can divide by two. Alternatively, assigning different contributions from distinct sections complicates the adjustment process.

The Definition of a Partition of Unity

To avoid discontinuities, we need to conceptualize a continuous version of our earlier approach, termed a partition of unity. This concept entails assigning portions of a function to each chart, ensuring that all segments sum to the whole function.

Formally, a partition of unity is a collection of functions mapping from a topological space X to [0, 1], meeting specific criteria at every point x.

Creating a Useful Partition of Unity

Although creating a partition of unity may seem complex, a straightforward method exists. By defining a new function for each chart that is positive within the chart and zero elsewhere, we can ensure continuity.

Too Many Charts

Introducing a partition of unity resolves certain challenges, but it raises another issue: summing an uncountable number of elements is impractical. Therefore, we assert that every point must be covered by a finite number of charts, leading to the concept of a locally finite open cover.

Locally Finite Open Cover

This term implies that every point in the space is included in at least one of our sets, which consist solely of open sets. The requirement of a locally finite open cover indicates that every point is encompassed by a finite number of open sets.

Charts Are Too Big

As previously noted, we should be able to make our charts smaller to minimize distortion. A space that always permits such a locally finite open cover is termed paracompact.

Still Too Many Charts

Despite limiting each point to a finite number of charts, we may still face the challenge of summing uncountably infinite elements. To avoid this, we stipulate that we can cover our manifold with a countable number of charts, thus defining it as second-countable.

Integration on a Manifold

To integrate over a manifold, we first establish a countable number of charts that collectively cover each point. Then, we create a useful partition of unity, multiply our function (f(x)) by each function in the partition, and integrate across the relevant charts. Finally, we aggregate our results.

Practical Integration Over a Manifold

While the theoretical integration process is beneficial, it can be cumbersome in practice. Nonetheless, theoretical frameworks remain essential for practical applications, such as simplifying integrals through the Generalized Stokes' Theorem.

The General Use of Partitions of Unity

In essence, partitions of unity enable us to extend local findings (from individual charts) into comprehensive conclusions applicable to the entire manifold.

The Formal Definition of Manifold

With our groundwork established, we can formally define a manifold as a second-countable Hausdorff space that is locally Euclidean. This definition is widely recognized, although some may relax the second-countability condition in favor of paracompactness. Ultimately, the goal remains the same: to ensure the presence of charts, unique limits, and partitions of unity.

The Manifolds We've Seen So Far

In closing, let's explore a few key manifolds of interest.

The Tangent Bundle

Every smooth manifold possesses a tangent bundle, which is itself a smooth manifold. This concept has been explored in depth in previous discussions, where the tangent space at a point represents all potential velocities at that location.

Riemannian Manifolds

A Riemannian metric is a form of a metric tensor, and every smooth manifold can be transformed into a Riemannian manifold. This structure consists of a real, smooth manifold equipped with a positive-definite inner product on the tangent space at each point.

Multiple Riemannian Metrics Mean Multiple Possible Riemannian Manifolds

Smooth manifolds can exhibit various Riemannian metrics, allowing different smooth manifolds to emerge from the same underlying structure through varying metric tensors.

The Cotangent Bundle

A cotangent space is dual to the covector space; by gluing all covector spaces together as done for the tangent bundle, we create the cotangent bundle, which is also a manifold.

Further Reading

For additional formal and alternative approaches to manifolds, consider the following resources:

• A quick playlist on manifolds from The Bright Side of Mathematics, aimed at proving the Generalized Stokes' Theorem.
• An article by Aidan Lytle that provides a formal take on manifolds.
• Various explanations on math.stackexchange and mathoverflow regarding manifold definitions and properties.

What's Next?

Having established a formal definition of manifolds, we will proceed to prove the Generalized Stokes' Theorem in the following article. This theorem allows for the transformation of certain integrals on manifolds into integrals over their boundaries, potentially simplifying problem-solving in various contexts.