Concept map for manifolds

Question

What exactly is manifold? What concepts do I need to learn in order to take on manifolds and concepts related to it?

Answer 1

jibounet has the right of it. Classically, a manifold is a set $M$ that locally "looks like" $\mathbb{R}^n$ (or a subset). For example, when you're thinking about some region of the Earth (your local state/province/district/what-have-you), you probably think of it as more-or-less flat (ignoring geographical features like mountains) as opposed to being a portion of something that is roughly spherical.

The same idea is behind manifolds. You have a set $M$ and, for a fixed positive $n$, a collection $\mathcal{A}$ of bijections $\varphi:U\to \mathbb{R}^n$, which I'll denote by $(U,\varphi)$, from subsets $U \subset M$ that together cover $M$, i.e. $$ M = \bigcup_{(U,\varphi) \in \mathcal{A}}U. $$ (I'm ignoring some technical conditions for the sake of emphasizing the concept. For example, you also need to assume that $(U,\varphi),(\psi,V) \in \mathcal{A}$ are somehow "compatible" on $U \cap V$ whenever $U\cap V \neq \emptyset$.)

You can then define a topology on $M$ based on the topology of $\mathbb{R}^n$ by saying that a set $S$ in $M$ is open if and only if $\varphi(S\cap U)$ us open for all $(U,\varphi)$. (More restrictions come up here. For example, you usually assume $M$ is Hausdorff and second countable.)

With more work, you can also define a differentiable structure on your manifold, from which you can define the notion of a $C^k$ function $f: M \to N$ between manifolds. [Edit: I've omitted my discussion here, since it was too hand wavy to be particularly helpful. See Alessandro's answer for a nice and brief definition of class $C^k$ differentiable manifolds.]

The prerequisites for an introductory course are multivariable calculus, linear algebra, and some basic topology. A course on differential equations (especially one covering systems of differential equations) would be useful, too. If you're looking for a good introductory text that still covers a good deal, I'd recommend Lee's Introduction to Smooth Manifolds.

Answer 2

You can approach manifolds in different ways, from different areas and of course at different degrees of deepness. I'll explain some of them.

(ANALYSIS) You can see a $k$-manifold essentially as a subset of an euclidean space $X\subseteq \mathbf{R}^n$ with the propriety of being locally diffeomorhpic to an open ball in $\mathbf{R}^k$. A diffeomorphism is a map $f:U\subseteq\mathbf{R}^n\to \mathbf{R}^k$ from an open set which is injective, differentiable and admits an inverse (restricting to image) that is differentiable.

(TOPOLOGY) The most abstract and neat definition comees from general topology, so in this case you are assumed to know point-set topology at a basic level. A (topological) $n$-manifold is a topological space $X$ endowed with an open cover $\{U_i\}_{i\in I}$ with the propriety that for each $i\in I$ there exist an open set $D_i\subseteq \mathbf{R}^n$ and an homeomorphism $\varphi : U_i\to D_i$. The couples $(\varphi_i, U_i)$ are called local coordinate charts and the set of all charts associated to an open cover is called an atlas for $X$; an atlas is called maximal when it contains every chart possible for $X$. In the aim of avoiding patological or trivial cases, we often require $X$ is second-countable and a Hausdorff space (or at least Fréchet).

The common definition used in modern geometry is the second, coming from topology. In fact, any other kind of manifold (differential, holomorphic, algebraic...) can be built up from this definition in that way. Let be $(U_j,\varphi_j),(U_k,\varphi_k)$ two coordinate charts for a topological manifold $X$. We call transition map the composite map $$\varphi_j\circ \varphi_k^{-1}:\varphi_k(U_j\cap U_k)\longrightarrow\varphi_j(U_j\cap U_k)$$

• A real differential manifold of class $\mathscr{C}^m$ is a topological manifold endowed with a maximal atlas having every transition map of class $\mathscr{C}^m$. Note that this is well defined, as transition maps act between open euclidean subsets.

• A complex holomorphic (resp. analitic) manifold is a topological manifold endowed with a maximal atlas having every transition map holomorphic (resp. analitic).

Algebraic varieties are manifold of particular kind, usually defined in other ways. But you can find also a definition of this flavour.