n-Dimensional Spheres as Manifolds

n-Dimensional Spheres as Manifolds

We will now begin to acknowledge some examples of manifolds. The first manifolds we will look at are the $n$-dimensional spheres which we define below.

Definition: The $n$-Dimensional (Unit) Sphere in $\mathbb{R}^{n+1}$ is the set of points $S^n = \{ (x_0, x_1, ..., x_n) \in \mathbb{R}^{n+1} : x_0^2 + x_1^2 + ... + x_n^2 = 1 \}$.

The $0$-dimensional unit sphere is $S^0 = \{ x_0 \in \mathbb{R} : x_0^2 = 1 \} = \{ -1, 1 \}$ and represents just two points on the real number line:

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The $1$-dimensional unit sphere is $S^1 = \{ (x_0, x_1) \in \mathbb{R}^2 : x_0^2 + x_1^2 = 1 \}$ and represents the circle centered at the origin with radius $1$.

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The $2$-dimensional unit sphere is $S^2 = \{ (x_0, x_1, x_2) \in \mathbb{R}^3 : x_0^2 + x_1^2 + x_2^2 = 1 \}$ and represents the sphere centered at the origin with radius $1$. It is important to note that $S^2$ is only the boundary of the closed ball centered at the origin with radius $1$.

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Locally, $S^n$ in $\mathbb{R}^{n+1}$ looks like $\mathbb{R}^n$. For example, if we were take a point $\mathbf{x}$ in $S^1$ and zoom in closed enough, locally, the part of $S^1$ close to $\mathbf{x}$ would look very much line a straight line (which the tangent line at $\mathbf{x}$ would approximate) . Thus it is not too hard to see that we could represent $S^1$ as a union of these straight one-dimensional lines to any desired degree of accuracy.

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