Circles and spheres as manifolds

1D Manifolds

A manifold is a space that is locally Euclidean.

A geometric object in $\mathbb{R}^{k}$ can be fit into $\mathbb{R}^{n}$ where n > k. (Fit a geometric object from a low dimensional space into a higher dimensional space).

An example of this is a 1D line segment that can fit into a 2D circle. If you take an infinitesimally small arc locally resemeles a 1D line segment.

Therefore a circle is an example 1D manifold embedded in a 2D space.

A more detailed definition of a manifold is a space that has continuous one-to-one and onto mappings whilst having space locally Euclidean.

As a result, any non-intersecting closed loop is an example of a manifold of a 1D line segment.

An intersecting closed loop such as an 8 because the crossing point does not locally resemble line segment

2D Manifolds

A sphere is an example of a 2D manifold embedded in a 3D space, as an infinitesimal small path of the surface of the sphere locally resembles a 2D Euclidean surface.

A slightly more formal look at Manifolds

Characteristics of manifolds

• Every point p of the manifold has an open neighborhood
• The mapping of the point p to the lower dimensional Euclidean space is referred to as a chart.
• The domain of the chart is the set U
• The image of the point p (mapping of p by the chart) is referred to as the local coordinates