Manifolds 46 | Example of a Manifold with Boundary

TL;DR

Manifolds with boundary allow charts to map into a half-space, so boundary points correspond to the half-space’s boundary.

Briefing Cornell Notes

Briefing

Manifolds with boundary are built by allowing coordinate charts to map into a “half-space,” which lets familiar manifold ideas extend cleanly to settings where Stokes’ theorem needs boundary terms. The key intuition comes from cutting an ordinary manifold into a piece: the cut edge becomes the boundary, and charts must respect that edge by mapping it to the boundary of the half-space.

The transcript starts by recalling the definition: a manifold with boundary uses the same smooth-manifold framework, except charts are permitted to land in a half-space. Concretely, the half-space is ^n restricted so that the first component is not positive (described as “the first component is not allowed to be positive,” i.e., one side of ^n). Charts are still homeomorphisms, but the target open sets live in the half-space with the subspace topology. This small change is what makes boundary geometry compatible with the rest of manifold theory.

To make the idea tangible, the example takes the 2-sphere S^2 and keeps only the Northern Hemisphere, defined as the points where the last coordinate x3 is greater than or equal to zero. The equator (where x3 = 0) remains part of the resulting space, so it should function as the boundary. However, a single chart is not enough: the boundary in the hemisphere is a circle, while the boundary of the half-space is a line extending to infinity. The solution is to restrict attention to a “quarter” of the hemisphere so the boundary becomes a half-circle that can match the half-space boundary under a homeomorphism.

A formal chart is then constructed by imposing an additional inequality, for instance x1 > 0, producing a region U on the hemisphere that corresponds to a quarter. The chart is obtained by a geometric simplification: rotate the picture so that the relevant coordinates align with the half-space directions, then project down to the half-space. In the described mapping H, three coordinates on U are sent to two coordinates in the half-space: the x2 coordinate is kept, while the x3 coordinate is flipped (the first coordinate on the target is given as min x3, i.e., the sign-adjusted x3). The transcript emphasizes that the boundary portion (where x3 = 0) lands on the boundary of the half-space, while the extra inequality (x1 > 0) ensures the image is an appropriate open set in the subspace topology.

By repeating the construction across all quarters, an atlas is assembled for the Northern Hemisphere as a manifold with boundary. The boundary of this manifold is identified with the equator, i.e., the set of points where x3 = 0, and it is a one-dimensional manifold without boundary—specifically S^1, a circle. The discussion closes by pointing toward the next step: extending tangent spaces and orientations to manifolds with boundary so that Stokes’ theorem can be formulated correctly in this broader setting.

Cornell Notes

Manifolds with boundary use the same chart-and-atlas framework as ordinary manifolds, except charts are allowed to map into a half-space (a subset of ^n with one coordinate restricted). The Northern Hemisphere of S^2 (x3 0) provides a concrete example: the equator (x3 = 0) becomes the boundary. A single chart cannot match the circle boundary to the half-space boundary line, so the hemisphere is split into quarters using an extra inequality like x1 > 0. After rotating and projecting, each quarter admits a homeomorphism into the half-space, and combining these gives an atlas. The resulting boundary is the equator, which is a 1D manifold without boundary, namely S^1.

Why does a manifold with boundary require charts into a half-space rather than all of ^n?

Because the boundary of the manifold must correspond to the boundary of the coordinate model. In the transcript’s definition, charts are homeomorphisms from open sets in the manifold to open sets in a half-space (a subspace of ^n where one coordinate is restricted). With the half-space’s subspace topology, the “edge” where the restricted coordinate hits its limit is mapped to the half-space boundary, making the boundary notion consistent across charts.

How does cutting the sphere produce a manifold with boundary?

Take the 2-sphere S^2 and keep only the Northern Hemisphere: points with x3 0. The equator (x3 = 0) remains inside the set, so it behaves like the boundary. This matches the general idea that the boundary is the cut edge of the original manifold.

Why can’t one chart describe the hemisphere with boundary?

The boundary of the hemisphere is a circle, but the boundary of the half-space is a line extending to infinity. A single homeomorphism from the whole hemisphere region to a half-space open set would not match these shapes. The transcript resolves this by restricting to smaller regions (quarters) so the boundary portion becomes a half-circle, which can be mapped to the half-space boundary line under a homeomorphism.

What inequalities define the quarter used for one chart?

Besides x3 0 (to stay in the Northern Hemisphere), an additional condition like x1 > 0 is imposed. This defines a region U that maps into the half-space in a way that turns the equator portion into a half-circle image. The extra inequality is crucial for getting the correct open-set behavior under the subspace topology.

How is the chart constructed from the quarter to the half-space?

The transcript describes rotating the figure so the relevant coordinates align with the half-space directions, then projecting down. The mapping keeps x2 and flips x3 via a sign change (the target’s first coordinate is given as min x3). Under this projection, points with x3 = 0 land on the half-space boundary, while the region defined by x1 > 0 ensures the image corresponds to an appropriate open set.

What is the boundary of the Northern Hemisphere in this example, and does it have its own boundary?

The boundary is the equator: the set of points where x3 = 0. The transcript notes this boundary is a one-dimensional manifold without boundary, identified with S^1 (a circle). Because it is a closed loop, it does not inherit a further boundary.

Review Questions

What changes in the definition of a manifold when introducing boundaries, and how does the half-space’s subspace topology matter?
In the Northern Hemisphere example, why is an atlas built from multiple quarter charts rather than a single global chart?
How does the equator become the boundary, and why is that boundary identified with S^1 rather than something with boundary?

Key Points

1
Manifolds with boundary allow charts to map into a half-space, so boundary points correspond to the half-space’s boundary.
2
A half-space model is treated with the subspace topology, ensuring chart images are open in the correct sense.
3
The Northern Hemisphere of S^2 (x3 0) is a manifold with boundary because the equator (x3 = 0) remains as the cut edge.
4
Matching the hemisphere’s circular boundary to the half-space’s line boundary requires splitting the hemisphere into smaller regions (quarters).
5
Each quarter chart is built by rotating and projecting so that x2 is preserved while x3 is sign-flipped in the target coordinates.
6
The boundary of the Northern Hemisphere is the equator, a one-dimensional manifold without boundary, namely S^1.
7
Extending tangent spaces and orientations to manifolds with boundary is necessary for formulating Stokes’ theorem correctly.

Highlights

The equator of the Northern Hemisphere (x3 = 0) becomes the boundary of the manifold with boundary.

A single chart fails because a circle boundary cannot be homeomorphic to the half-space boundary line; quarter charts fix this.

The boundary of the example is S^1, and it has no boundary itself.

Charts into a half-space are still homeomorphisms, but the target open sets live in the half-space’s subspace topology.

Topics

Manifolds With Boundary
Half-Space Charts
Northern Hemisphere
Equator Boundary
Stokes Theorem