Notation of sphere in $\Bbb{R}^n$.

Question

In $\Bbb{R}^n$, the sphere centred at $x\in \Bbb{R}^n$ with radius $l$ is called $S^{n-1}(x,l)$ or an $(n-1)$-sphere.

Why do we called it an $(n-1)$-sphere? It's certainly not defined on $\Bbb{R}^{n-1}$ instead of $\Bbb{R}^n$.

Thanks in advance!

People are all saying more or less the same thing in different ways. The $n-1$ refers to the dimension of the sphere itself, rather than the dimension of the space in which the sphere lives. If instead of talking about points with distance precisely $l$ from $x$, you talked about points having distance $\leq l$ from $x$, you'd call that an $n$-dimensional disc. — Ryan Budney, Aug 27 '13 at 04:08

score 6 · Accepted Answer · answered Aug 27 '13 at 04:00

6

Since the sphere is defined as the set of point on the sphere and not inside it, a sphere in $\mathbb{R}^n$ is just a $n-1$ dimensional surface.

Another way of looking at this is that every point on the sphere with a given radius can be expressed by $n-1$ angles.

answered Aug 27 '13 at 04:00

Nathaniel Bubis

33,018

1

$n-1$ local coordinates in general (angular or not) – obataku Aug 27 '13 at 05:04
@oldrinb - for the sphere with a given radius, the common characterization involves $n-1$ angles. Of course you can invent other ones, but I find the angles representation gives some helpful intuition. – Nathaniel Bubis Aug 27 '13 at 05:12
yes @nbubis the angles correspond to arcs of curvilinear coordinate axes thoush yes? – obataku Aug 27 '13 at 05:12
@oldrinb - The wiki page describes the angles under the section "Spherical coordinates". – Nathaniel Bubis Aug 27 '13 at 05:18
yes @nbubis I am familiar, but my point was that the angles themselves are just like 'arcs' on circular coordinate axes :p – obataku Aug 27 '13 at 18:51

score 1 · Answer 2 · answered Aug 27 '13 at 04:00

1

Locally, the sphere looks like $\Bbb{R}^{n - 1}$, and the $(n-1)$-sphere is an $(n - 1)$-dimensional manifold. The ball, on the other hand, is an $n$-manifold.

answered Aug 27 '13 at 04:00

score 1 · Answer 3 · answered Aug 27 '13 at 04:00

As a manifold, the $(n-1)$-sphere has dimension $n - 1$. This means that close up, any small part of the $(n-1)$-sphere will look like $\Bbb R^{n-1}$. To be precise, every point $x \in S^{n-1}$ has an open neighborhood that is homeomorphic to $\Bbb R^{n-1}$.

Notation of sphere in $\Bbb{R}^n$.

3 Answers3