Is the Surface of a Cube a Differential Manifold?

The question of whether the surface of a cube is a differential manifold has been a point of interest in the field of differential geometry. To answer this, it is essential to understand the definition of a differential manifold and analyze the key characteristics of the cube's surface.

Definition and Key Points

A differential manifold is a topological space that is locally similar to Euclidean space and has a differentiable structure. Specifically, for the surface of a cube, we can view it as a 2-dimensional object in 3-dimensional space, which is a 2-dimensional manifold with boundary.

Local Structure

Each face of the cube is a square that is homeomorphic to an open subset of ?2. Around the edges and corners, the surface can also be locally approximated by Euclidean spaces.

Chart and Atlas

We can cover the surface of the cube with charts, where each chart corresponds to a face of the cube or a region around an edge or vertex. The transitions between these charts are differentiable where they overlap.

Boundary

The surface of the cube has edges that act as boundaries where the local structure may not be entirely smooth. Despite this, the surface still qualifies as a manifold with boundary.

Conclusion

In summary, the surface of a cube can be considered a 2-dimensional differential manifold with boundary. The cube's surface meets the criteria for a manifold where it is locally Euclidean and the transition functions are differentiable.

Discussion on Kinks and Non-differentiability

It has been noted that not all surfaces are differential manifolds, especially those that have kinks. The vertices of a cube, which are kinks, prevent the surface from being differentiable. This is essentially the same reason as taught in first-year Calculus: a kink does not allow for a well-defined tangent space.

Consider a square as a subspace of the plane. In this case, the square is not a submanifold of the plane since there are no subspace charts that can make the square into a manifold. The corner of the square is particularly problematic, as it does not have a well-defined tangent space. Two candidate spaces are present: one vertical and one horizontal. However, two diffeomorphic spaces must have isomorphic tangent spaces at each point, and no subspace chart can be diffeomorphic to ?1 because every point in it has a well-defined 1-dimensional tangent space, while the corner of the square does not have a well-defined tangent space.

Isomorphism and Tangent Spaces

This violates the condition that a diffeomorphism between spaces gives rise to an isomorphism between the respective tangent spaces. The induced map or pushforward is the key concept here. Therefore, a square cannot be a submanifold of the plane as defined by differential geometry.

However, as pointed out by David Joyce, a square can indeed be given manifold charts. It is possible to use the result that if X -transform is a topological space that is homeomorphic to the space Y transform, and Y transform is a smooth manifold, then X transform can be made into a smooth manifold by "pulling back" the charts along the homeomorphism between the two spaces. Here, the surface of the cube (Xtransform) can be made into a smooth manifold by pulling back the charts from a sphere (Ytransform).

In conclusion, while a cube's surface can be analyzed as a differential manifold with boundary, it is not a submanifold of the plane due to the presence of kinks and non-differentiability at the vertices. The surface can, however, be given a manifold structure by using appropriate charts.