For the n-sphere, and also when taken together with a product of open intervals, we have the following.

This term applies to opposite points on a circle or any n-sphere.

For example, the n-sphere in R is called a hypersphere.

Non-trivial means that neither of the two is an n-sphere.

Thus, the n-sphere centred at the origin is defined by:

In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to .

It can be thought of as the negative-curvature analogue of the n-sphere.

Example: The oriented n-manifolds have an addition operation given by connected sum, with 0 the n-sphere.

The n-sphere is the double of the n-ball.

The group of smooth structures of an oriented PL n-sphere.

Also is the Paneitz operator, , on the n-sphere.

For example, the circle (and more generally, n-sphere) are null-cobordant since they bound an (n+1)-disk.

The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map.

The original proof of the sphere theorem did not conclude that M was necessarily diffeomorphic to the n-sphere.

The n-sphere S is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions.

It thus has the same homotopy groups and the same homology groups, as the n-sphere.

The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold.

The space Z is therefore isomorphic to the projectivized cotangent bundle of the n-sphere.

Consider the standard CW-decomposition of the n-sphere, with one zero cell and a single n-cell.

An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor.

A common construction in homotopy is to identify all of the points along the equator of an n-sphere .

The group of conformal transformations of the n-sphere is generated by the inversions in circles.

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension.

The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd.

The set of vectors in R whose Euclidean norm is a given positive constant forms an n-sphere.