^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] | [[Schutz - 2.2 - The Sphere as a Manifold|2.2 The Sphere as a Manifold]]>> # Definition of a Manifold ^2c5b55 > **Manifold** > A set $M$ is defined to be a *manifold* if each point of *M* has an [[Schutz - 1.1 - The Space R^n and its Topology#^224639|open]] [[Schutz - 1.1 - The Space R^n and its Topology#^37f45e|neighborhood]] which has a [[Schutz - 1.2 - Mappings#^1aee5f|continuous]] [[Schutz - 1.2 - Mappings#^6c8b58|bijection]] onto an open set of $\mathbb{R}^n$ for some $n$. The dimension of the manifold is then $n$. > More simply, it means that $M$ locally "looks like" $\mathbb{R}^n$. ^31cc42 The definition has to involve only open sets and not the whole of $M$ and $\mathbb{R}^n$ because we do not want to affect the global topology of $M$, as we will see in [[Schutz - 2.2 - The Sphere as a Manifold|§2.2]]. Moreover, the map is only required to be bijective, not preserve any geometrical notions. Given a point $P$ on a manifold $M$, the aforementioned bijection assigns to $P$ an $n$-tuple of numbers $(x^1(P), x^2(P), \ldots x^n(P)) = x^i(P)$. These numbers are called the coordinates of $P$ under the map. See the figure below. ![[Pasted image 20210123175907.png]] Another way of thinking about an $n$-dimensional manifold is as a set which can be given $n$-independent coordinates in some neighborhood of any point, since these coordinates actually define the bijection discussed above. > **Coordinate Chart** > A *coordinate chart*, or simply a *chart*, is an open neighborhood $U$ of a point $P\in M$ with the bijection $f:U\rightarrow \mathbb{R}^n$. That is, the chart is $(U, f)$. ^7b8801 Note that in the definition of a chart above, it is important to observe that the neighborhood $U$ does not necessarily include all of $M$, and in general it can't. In general, each neighborhood has its own map (so that a manifold is covered by multiple charts), and each point in $M$ must lie in at least one such neighborhood. It is easy to see that these neighborhoods (since they are open by definition, see [[Schutz - 1.1 - The Space R^n and its Topology#^37f45e|here]]) must overlap so that they can cover the whole manifold. >**Atlas** > An atlas is a collection of charts covering the whole manifold > **Coordinate Transformation** > Suppose $U$ and $V$ are overlapping neighborhoods (i.e. $U \cap V \neq \emptyset$). Their intersection is open by definition. Suppose we have the charts $(U, f)$ and $(V, g)$. Note that $f(U) \subset \mathbb{R}^n$ may be completely distinct from $g(V) \subset \mathbb{R}^n$. Thus, there is some equation relating these coordinate systems. > Pick a point in the image of the overlap under $f$ and call it $x^i$ as shown in the figures below. Since $f$ is a bijection, it has an inverse $f^{-1}$ that maps to a unique point in the overlap, $f^{-1}(x^i) = S$. Now take a look at the image of $S$ under $g(V)$, call it $y^i$, i.e. $g(S) = g(f^{-1}(x^j)) = y^i,\, i, j = 1 \ldots n$. In this way, we obtain a *coordinate transformation* > $y^i = y^i(x^j)$ > Or in a more verbose manner: > $\begin{align} y^1 &= y^1(x^1, x^2, \ldots x^n) \\ y^2 &= y^2(x^1, x^2, \ldots x^n)\\ &\vdots \\ y^n &= y^n(x^1, x^2, \ldots x^n)\\ \end{align}$ ^c15345 ![[Pasted image 20210123181322.png]] ![[Pasted image 20210123181337.png]] If the partial derivatives of order $k$ or less of all the $\{y^i\}$ with respect to all the $\{x^i\}$ exist and are continuous, then the charts $(U, f)$ and $(V, g)$ are said to be *$C^k$-related*. If it is possible to construct an atlas in such a way that every point point of $M$ where every chart is $C^k$-related to every other one it overlaps with, then the manifold $M$ is said to be a $C^k$-manifold. ^ddceb2 > **Differentiable Manifold** > A *differentiable manifold* is a manifold of class $C^1$. ^d7ff88 In most physics applications, we will assume a $C^\infty$ manifold, but sometimes it will be useful to assume a $C^\omega$ manifold, as discussed in [[Schutz - 1.3 - Real Analysis#^4bdffa|§1.3]]